The most beautiful equation in math, explained visually [Euler’s Formula]

The most beautiful equation in math, explained visually [Euler’s Formula] Welch Labs 572K subscribers Subscribe 26K Share Download Clip Save 838,357 views Aug 12, 2024 Welch Labs Imaginary Numbers Book Pre-Order https://www.welchlabs.com/resources/i... Book Digital Version https://www.welchlabs.com/resources/i... Euler’s Formula Poster! https://www.welchlabs.com/resources/e... Poster Digital Version https://www.welchlabs.com/resources/e... Special thanks to the Patrons: Juan Benet, Ross Hanson, Yan Babitski, AJ Englehardt, Alvin Khaled, Eduardo Barraza, Hitoshi Yamauchi, Jaewon Jung, Mrgoodlight, Shinichi Hayashi, Sid Sarasvati, Dominic Beaumont, Shannon Prater, Ubiquity Ventures, Matias Forti Tattoo by @themutemaker - thank you! Check out her awesome work here: / themutemaker Welch Labs Ad free videos and exclusive perks: / welchlabs Watch on TikTok: / welchlabs Learn More or Contact: https://www.welchlabs.com/ Instagram: / welchlabs X: / welchlabs References & Notes Welch Labs Imaginary Numbers Series: • Imaginary Numbers Are Real [Part 1: I... Excellent History of Logarithms by Florian Cajori Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 1, 1913, pp. 5–14. *JSTOR*, https://doi.org/10.2307/2973509 . Accessed 22 July 2024. Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 2, 1913, pp. 35–47. *JSTOR*, https://doi.org/10.2307/2974078 . Accessed 22 July 2024. Cajori, Florian. “History of the Exponential and Logarithmic Concepts:” *The American Mathematical Monthly*, vol. 20, no. 3, 1913, pp. 75–84. *JSTOR*, https://doi.org/10.2307/2973441 . Accessed 22 July 2024. Cajori, Florian. “History of the Exponential and Logarithmic Concepts.” *The American Mathematical Monthly*, vol. 20, no. 4, 1913, pp. 107–17. *JSTOR*, https://doi.org/10.2307/2972960 . Accessed 22 July 2024. Nice History of Euler’s Formula Sandifer, Ed. e, pi and i: Why is “Euler” in the Euler identity? http://eulerarchive.maa.org/hedi/HEDI... Much of the visual approach presented here comes from Needham’s incredible book: Needham, T. (1997). Visual Complex Analysis. United Kingdom: Clarendon Press. Other books referenced Maor, E. (2011). E: The Story of a Number. Ukraine: Princeton University Press. Penrose, R. (2021). The Road to Reality: A Complete Guide to the Laws of the Universe. United Kingdom: Knopf Doubleday Publishing Group. Dunham, W. (2022). Euler: The Master of Us All. United States: AMM Press. Wilson, R. (2019). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. United Kingdom: Oxford University Press. Nahin, P. J. (2010). An Imaginary Tale: The Story of √-1. Ukraine: Princeton University Press. Stillwell, J. (2013). Mathematics and Its History. United Kingdom: Springer New York. Euler’s Amazing 1747 Paper Euler, Leonard. "Sur les logarithmes des nombres négatifs et imaginaires” Written in 1747, but not published until 1862. Euler did publish a similar paper in 1749. See Cajori #3 above. English Translation: https://scholarlycommons.pacific.edu/... Note on Benroulli’s area of sectors Euler’s counterexample using Bernoulli’s sector area comes in a couple flavors. The one presented in his 1747 paper "Sur les logarithmes des nombres négatifs et imaginaires” is a bit different than an earlier example in a letter to Bernoulli. I chose the earlier example for clarity. See Cajori vol 2 and Sandifer. Feynman Lectures - Algebra https://www.feynmanlectures.caltech.e... Transcript Follow along using the transcript. Show transcript Transcript Search in video 0:00 there's this really beautiful idea in 0:01 math that many of the equations and 0:04 graphs we spend our time with are just 0:05 Shadows of a more elegant powerful and 0:08 higher dimensional mathematics this 0:11 higher dimensional mathematics has 0:12 everything to do with the imaginary 0:14 number I the square root of minus1 0:18 extending polinomial like x^2 + 1 to 0:20 include imaginary numbers is relatively 0:23 straightforward but there's one pair of 0:25 functions exponentials and logarithms 0:28 where the application of imaginary 0:29 numbers is far less clear 2 to ^ of 3 0:33 means we should multiply two by itself 0:35 three times but what could it possibly 0:37 mean to raise two to the power of the 0:39 square < TK of 0:41 minus1 in the early 1700s disagreements 0:44 over these questions threaten the very 0:46 foundations of mathematical analysis 0:48 until Leonard Oiler stepped in with a 0:50 solution so surprising and elegant that 0:53 it's often called the most beautiful 0:54 equation in math this video is sponsored 0:57 by me I've written a whole book on 1:00 imaginary numbers it's coming out later 1:02 this year you can pre-order a copy using 1:03 the link in the description below on 1:06 March 16th 1712 godfried Wilhelm libnet 1:10 co-founder of calculus sent this letter 1:12 to the mathematician Johan beri where he 1:15 claimed that it was impossible to take 1:16 the logarithm of a negative number as 1:19 liet would detail in a later letter to 1:21 berui logs are just another way to write 1:23 exponents and simply don't make sense 1:25 for negatives log base 2 of 8 is three 1:29 because 2 to the 3 power is 8 log base 2 1:32 of 1 is 0 because 2 the^ of 0 is 1 the 1:36 reason 2 the 0 is 1 and not zero or 1:39 something else is that this definition 1:41 preserves an important property of 1:43 exponents and logarithms when we 1:45 multiply two terms with the same base 1:47 their power is add 2^ 2 * 2 cubed 1:50 results in five total multiplications of 1:52 2 or 2 5th defining 2 the 0 as 1 1:55 preserves this property 2 3 + 0 = 2 3r * 2:00 2 the 0 if and only if we Define 2 the 0 2:03 as 1 as we'll see Finding mathematically 2:06 consistent ways to handle logs and 2:08 powers of various types of numbers is 2:10 critical to making sense of Oilers 2:12 approach finding a sensible definition 2:14 for logarithms of fractions between 0 2:17 and 1 is also relatively straightforward 2:20 dividing exponential terms with the same 2:22 base is equivalent to subtracting their 2:24 exponents 2 Cub / 2^2 is equal to 2 3 - 2:29 2 this means that 2 Theus 2 on its own 2:32 must equal 1/ 2^2 this is how we arrive 2:35 at the logarithms of fractions the log 2:37 base 2 of 1/4 is equal to -2 because 2 2:40 to -2 is 1/4 note that we've broken away 2:44 from the simple definition of iner 2:46 exponents where an exponent of in means 2:48 we multiply a number by itself in times 2:51 this doesn't make sense for negative 2:53 exponents however our fractional 2:55 definition nicely preserves our 2:56 properties and when we plot the value of 2:58 log base 2 for num is greater than one 3:00 for one and for fractions we get a nice 3:02 continuous curve now when we finally 3:05 reach the logarithms of negatives as 3:07 liet's claims we basically run out of 3:09 options what power of two would result 3:12 in minus one clearly positive numbers 0 3:15 and negatives will not work in a 3:18 response letter a few weeks later beri 3:20 flatly rejects libet's claim beri argues 3:23 that since the derivatives of log X and 3:25 log ofx both evaluate to 1 /x log of x 3:29 and log of negative X are equal this 3:32 would mean that the log function wasn't 3:33 just one curve but actually had a 3:35 positive and negative branch that are 3:37 symmetric across the y AIS and the log 3:39 of minus one just equals the log of one 3:42 so zero liit was not convinced and the 3:45 mathematicians went back and forth for 3:47 over a year without making progress 3:49 towards a meaningful resolution when 3:51 Leonard Oiler a student of beri picked 3:54 up this still unresolved problem a few 3:56 decades later he wrote that not finding 3:58 a Reconciliation would be an indelible 4:00 stain on analysis at this point calculus 4:03 and Analysis were still only five or six 4:05 decades old and relied on some serious 4:08 hand waving to handle the infinitely 4:10 large and infinitely small quantities 4:12 required Oilers speculated that other 4:15 mathematicians had encountered the same 4:16 difficulties with logarithms but had not 4:19 published out of fear of breaking this 4:21 new branch of math without being able to 4:23 fix it in an incredible 133-page paper 4:26 written in 1747 Oiler points out logical 4:29 flaw on with the solutions of both 4:30 bernui and liet and lays out an 4:33 unbelievably elegant solution that 4:35 remains largely unchanged today first 4:38 Oiler demolishes his teacher B's theory 4:41 that log of x equals log ofx Oiler 4:44 argues that just because log X and log 4:46 of negative X have the same derivative 4:48 this does not mean that the functions 4:49 themselves are equal the same faulty 4:52 logic can be used to show that log of 2x 4:54 = log of x which is clearly false from 4:57 here Oiler draws on an example from ber 5:00 himself to show that Li nits is 5:01 partially correct in 1702 while 5:04 investigating the areas of sectors of 5:06 circles beri got stuck on this difficult 5:09 integral that computes the area of a 5:11 sector as a function of the position XY 5:13 on the corner of the sector beri 5:16 realized that if he effectively pulled a 5:18 square root of minus one out of his 5:20 expression he was able to evaluate the 5:23 integral the result is a fairly complex 5:25 expression for the area of the sector 5:28 involving I and a log 5:30 Oiler showed that if he chose the point 5:32 XY to be 01 br's equation would simplify 5:36 into the logarithm of1 ided 4 I Oiler 5:40 argued that since the area of the sector 5:42 was clearly a real quantity that the i 5:44 in the denominator would have to cancel 5:46 with an i in the numerator meaning that 5:48 the log of1 had to be imaginary however 5:52 just declaring that logarithms of 5:53 negative numbers are imaginary does not 5:55 fix the problem and as Oiler goes on to 5:57 show it leads to a new set of 5:59 contradictions 6:00 a key property of logarithms is that 6:02 raising the argument of a logarithm to a 6:04 power is equivalent to multiplying the 6:06 logarithm by the power the log of 2 6:09 cubed is equal to 3 * the log of 2 6:12 having established that the logarithm of 6:13 minus1 is imaginary we can write log of 6:16 minus1 = B * I where B is some real 6:19 number multiple of I Oiler then 6:22 considered the case where the ne -1 6:24 inside the logarithm is raised to the^ 6:26 of two applying our property this should 6:29 be equal to 2 * the logarithm of1 or 6:32 2bi however since - 1^2 is just 1 and we 6:36 know that the logarithm of 1 is zero it 6:39 also appears that the log of -1 s is 6:41 equal to 0 which we just showed cannot 6:44 be the case oer was stumped fully 6:48 appreciating that the foundations of 6:49 calculus were at stake Oiler wrote that 6:52 he was tormented by these difficulties 6:54 and that for a long time it seemed 6:56 impossible to rescue the truth finally 6:59 oil found a way forward what if both 7:02 answers Zer and 2bi were correct what if 7:06 logarithms didn't have a single answer 7:08 but many maybe even an infinite number 7:10 of 7:11 solutions to see how this is possible we 7:14 need to return to the very first 7:16 question we considered what does it mean 7:18 to raise a number to the power of the 7:19 square root of minus1 although Oiler 7:22 solution to this problem is incredibly 7:24 elegant his path to the solution is not 7:27 the most intuitive Oiler put a huge 7:29 amount of trust into his symbols and was 7:32 absolutely masterful at finding creative 7:33 ways of manipulating equations oil's 7:36 mathematical style and a general 7:38 mistrust among mathematicians of 7:40 imaginary numbers meant that although 7:42 Oiler 1747 solution is essentially 7:44 correct by modern standards it would 7:46 take over 50 years well after oil's 7:49 death to be widely accepted a key factor 7:52 to the broader acceptance of Oiler 7:54 solution was the development of more 7:56 visual ways of thinking about imaginary 7:58 numbers let's use this more visual 8:01 intuitive approach to arrive at Oiler 8:03 answer and see how he saved calculus 8:06 imaginary numbers are typically used as 8:08 parts of complex numbers which have real 8:11 and imaginary components often written 8:13 as a plus bi I we can visualize complex 8:16 numbers as points on the complex plane 8:18 where the real axis is horizontal and 8:20 the imaginary axis is vertical we can 8:23 multiply complex numbers together using 8:24 the distributive property and the 8:26 product of two complex numbers is 8:28 generally another complex number a 8:30 notable exception is when we multiply a 8:32 complex number by its conjugate this 8:34 just means flipping the sign of the 8:36 imaginary terms when we work out the 8:38 multiplication in this case the 8:40 imaginary parts will cancel taking the 8:42 square root of this result gives the 8:44 distance between the number and the 8:46 origin on the complex plane also known 8:48 as the magnitude of the number finally 8:51 it's very helpful sometimes to express 8:53 complex numbers in polar form where 8:56 instead of writing the real and 8:57 imaginary parts we give the magnitude 8:59 and the the angle measured from the xais 9:02 multiplying complex numbers in polar 9:03 form is particularly useful because it 9:06 turns out that all the distributive 9:07 property algebra is completely 9:09 equivalent to multiplying together the 9:11 magnitudes and adding the angles of our 9:13 numbers so when we multiply two complex 9:15 numbers together their magnitudes 9:17 multiply and their angles add now 9:19 returning to our problem of what it 9:21 means to raise a number to an imaginary 9:23 power let's consider 2 to the power of 9:25 some number B * I an easy operation we 9:29 can try right away is multiplying this 9:31 expression by its complex conjugate 2 9:33 the minus bi which should give us the 9:36 distance squared between the origin and 9:38 2bi following our additive exponent 9:40 property the exponents cancel and we're 9:43 left with 2 the 0al one this is a 9:47 powerful result since our result is a 9:50 constant that doesn't depend on B it 9:52 means that the magnitude or distance 9:54 from the origin of 2 to the bi is 9:57 constant so 2 to the bi Must Fall 9:59 somewhere on a circle of radius one 10:02 perhaps most surprising here is that 10:04 unlike functions of real exponents which 10:06 grow and grow as their arguments 10:07 increase functions with imaginary 10:09 exponents are completely confined to the 10:11 unit circle as we'll see this has 10:14 everything to do with Oilers assertion 10:15 that logarithms have an infinite number 10:17 of solutions now the missing piece of 10:20 the puzzle of course is where exactly 10:22 does 2 to thebi fall on the unit circle 10:25 we now know that the magnitude of 2 to 10:27 thebi is one but what is its angle to be 10:30 more mathematically precise we're saying 10:32 that 2 to the bi equals some point on 10:34 the unit circle let's say it's at angle 10:37 Theta so we can write it as a complex 10:39 number with cosine Theta for the real 10:41 part and sin Theta for the imaginary 10:43 part now can we figure out how B and 10:46 Theta are connected if we plug in Bal 1 10:49 what happens to Theta if we can figure 10:51 this out we will have completely solved 10:53 our problem to get to the bottom of this 10:55 we need to go one level deeper into how 10:58 exponential functions 10:59 really work as we've seen It's 11:01 relatively straightforward to make sense 11:03 of integer exponents regardless of if 11:05 these exponents are positive Zer or even 11:07 negative but what about fractional or 11:10 decimal exponents simple fractional 11:12 exponents like x to the 1/2 are pretty 11:14 straightforward by preserving the 11:16 property that powers of powers multiply 11:19 we can show that x to the 1/2 is 11:20 equivalent to the square root of x but 11:23 what about more complicated powers like 11:26 0.587 sure you can plug this into your 11:28 calculator but how does it really work 11:31 the first person to really attack this 11:32 problem was Henry Briggs in 1617 when 11:35 constructing is incredibly prolific 11:37 tables of base 10 logarithms Briggs 11:40 found a clever computational approach 11:42 that works by basically zooming way in 11:44 on an exponential curve until it looks 11:46 like a straight line starting with an x 11:48 value of one for example we can zoom 11:51 into our curve by dividing X by two and 11:53 Computing the corresponding y value in 11:56 this case 2 ^ of 1/2 equal theare < TK 11:59 of 2 2 or roughly 12:00 1.414 repeating this operation again to 12:03 zoom in further we reach x = 1/4 and Y = 12:06 the < TK of the square root of two which 12:08 is equivalent to the 4th root of two or 12:10 about 12:11 1.89 after zooming in more and more 12:14 Briggs remarkably took 54 successive 12:16 square roots to the 33rd decimal place 12:19 we reach a tiny neighborhood where 2 to 12:21 the x is very nearly equal to 1 + Alpha 12:25 * X this is the equation for a straight 12:27 line in this neighborhood with a slope 12:29 of alpha Briggs used a slightly 12:31 different formulation but in our example 12:33 here the slope Alpha comes out to about 12:36 0.693 so when X is close to 0er 2 to the 12:40 x is very nearly equal to 1 + 0.693 * X 12:45 we can now use this fact as Briggs did 12:47 to solve arbitrary logarithm or 12:49 exponential problems like 2 to the power 12:51 of 12:52 0.587 taking our X starting value is 12:55 0.587 we first map to our tiny 12:58 neighborhood by either taking successive 13:00 divisions by two as Briggs did or we can 13:02 go in one big step to zoom in by a 13:05 factor of a th for example we divide 13:07 0.587 by a th000 so 13:11 0.587 assuming we've made it to our tiny 13:14 neighborhood we can now find 2 to the 13:16 power of 13:18 0.587 using our simple linear formula so 13:22 1 + 13:23 0.693 * 13:26 0.587 gives 13:28 1.004 13:30 46791 now that we know 2 to the power of 13:34 0.587 we can zoom back out to get our 13:36 final answer by raising this result to 13:38 the power of a th Computing about 13:42 15019 plugging this problem into a 13:44 modern calculator gives an answer a 13:46 little closer to 13:47 1502 this is just because a factor of a 13:50 th000 isn't really zooming in that far 13:52 if we zoom in by a factor of a million 13:54 instead our answers match to the sixth 13:57 decimal place we can wrap up what we're 13:59 doing here nicely into a single equation 14:02 2 to X approximately equals our linear 14:05 equation 1 + 0.693 * x divided by our 14:09 zooming in Factor n all to the power of 14:12 n to zoom back out the bigger n is the 14:15 more precise our answer is injecting a 14:18 little calculus 2 to the x is exactly 14:21 equal to the Limit as n approaches 14:22 Infinity of our expression this is still 14:25 not quite correct though because we 14:27 rounded the slope of our line Alpha is 14:30 0.693 the way this is typically handled 14:32 is to make the slope term equal to one 14:34 by changing our base if we Define a new 14:37 variable x = z / 0.693 and substitute 14:42 we're left with a simpler expression on 14:43 the right and a new base of 2 the Power 14:46 of 1 over 0.693 on the left given more 14:50 decimals of precision for Alpha our new 14:52 base of valuates to 14:55 2.71828 also known as the mathematical 14:57 constant e putting all this together we 15:00 have an equation that is often used as 15:02 the definition of the exponential 15:03 function e to the Z equals the limit as 15:06 in approaches Infinity of 1+ z/ n all to 15:09 the power of n this equation elegantly 15:12 describes the zooming in linearizing and 15:14 zooming out procedure that Henry Briggs 15:16 developed to compute the exponents and 15:18 logarithms of decimals and fractions now 15:21 that we have a deeper foundation for 15:23 what an exponential function really is 15:25 we're finally ready to make our final 15:27 attack on imaginary exponent 15:29 the key idea from here on out is 15:31 figuring out if Brigg's infinite zooming 15:33 in definition of exponentials works with 15:36 imaginary numbers and if so where on the 15:38 unit circle a number to a specific 15:40 imaginary power will land so what 15:43 happens if we plug in theun of1 into 15:46 Brigg's equation can we trust the 15:49 results even though Brigg's equation was 15:51 derived using real 15:52 numbers what does all of this mean 15:55 geometrically let's plug in Z equals I 15:58 and temporarily drop the limit and use a 16:00 smaller value of n let's say six this 16:03 will reduce the Precision of our result 16:05 but the intuition should still hold so 16:07 we have e to the power of I 16:09 approximately equals 1 + I over 6 all to 16:12 the 6th power 1 + I over 6 is not too 16:15 tricky to think about it's just a 16:17 complex number with a real part of one 16:19 and an imaginary part of 1 over 6 brig's 16:22 formula is telling us to multiply this 16:24 complex number by itself six times 16:27 performing the first multiplication we 16:29 get 35 over 36 + I over 3 the polar form 16:33 of complex numbers is helpful here as we 16:34 continue multiplying we can write 1+ I 16:37 over 6 as a complex number with a 16:39 magnitude of 1.014 and an angle of 9.46 16:43 De since multiplying complex numbers in 16:46 polar form is just a matter of 16:47 multiplying magnitudes and adding angles 16:50 we can jump to our final answer by 16:52 raising 1.014 to the 6th power and 16:55 multiplying 9.46 de by 6 giving a final 16:59 magnitude of 1.86 and an angle of 56.7 17:03 74° note that our answer is not quite on 17:06 the unit circle as expected but this is 17:08 just because we're using a fairly small 17:10 value for n if for example we bump into 17:13 a th we land much closer to the unit 17:16 circle so could this result be correct 17:20 is it consistent with all the nice 17:21 exponential and logarithm properties 17:23 that we've seen thus far and can it help 17:25 us figure out the relationship between B 17:27 and Theta in the unit circle equation we 17:30 derived earlier for 2 the Power of bi 17:33 let's go ahead and rewrite this equation 17:35 in terms of the base e to better line up 17:37 with Brigg's equation since B is just an 17:40 arbitrary constant we can introduce a 17:42 new variable B is equal to C / 17:45 0.693 just as we did with Brigg's 17:47 equation substituting we can rewrite our 17:49 equation as e to the CI is equal to 17:52 cosine theta plus I sin Theta and our 17:54 job now is to find the connection 17:56 between C and Theta when we plugged I 17:58 into Brigg's equation effectively using 18:00 a c value of one and used an N value of 18:03 a th our result was a complex number 18:06 with a magnitude of 18:08 1.005 and an angle of 5729 6° this angle 18:12 should be equivalent to the Theta value 18:14 in our coine plus I sin Theta equation 18:16 when C equals 1 so when C equals 1 theta 18:20 equals about 5729 18:22 6° let's plug in more values of c into 18:25 Brigg's equation and see how Theta 18:27 changes plugging in the values 0 through 18:30 5 for c a pattern emerges for Theta with 18:33 Theta increasing by a constant value of 18:36 about 5729 6° for each increase of C by 18:39 1 so this result is telling us that if 18:42 we raise e to the imaginary powers of I 18:44 2 I 3 I and so on our answer moves 18:47 around the unit circle in equal steps of 18:50 about 5729 6° now if we make a small 18:54 change and measure Theta in radians 18:56 instead of degrees we see something 18:59 truly remarkable each increase of C by 1 19:03 moves Theta by 180 / Pi or exactly one 19:08 radian so if we measure Theta in radians 19:11 instead of in degrees the value we've 19:13 been calling C is exactly equal to Theta 19:17 finally we've arrived at Oiler 19:18 incredible formula e ^ of I theta equals 19:22 cosine of theta plus I sin 19:25 Theta now before we celebrate too much 19:28 we never really checked that Brigg's 19:29 formula makes sense when our inputs are 19:31 imaginary after all Briggs developed his 19:34 formula by zooming in on a very real 19:36 exponential curve that has nothing to do 19:38 with imaginary numbers when we plugged 19:40 in I into brig's formula using Nal 6 we 19:44 saw that visually this looks like six 19:46 thin triangles stacked together to form 19:48 an overall angle each triangle has an 19:50 angle of 9.46 De and when we multiply 19:54 this number by itself six times these 19:56 angles are added together to produce a 19:58 fin angle of about 57° or one radian as 20:02 we increase in we get more and thinner 20:04 triangles and a more accurate estimate 20:07 of e to the power of I considering our 20:09 first thin triangle for a moment if n is 20:12 100 for example then the base of our 20:14 triangle is equal to 1 the real part of 20:17 1 plus I over 100 and the height of our 20:19 triangle is equal to the imaginary part 20:22 so 1 over 100 now where it gets 20:24 interesting is if we consider the 20:26 connection between the small angle of 20:28 this thin triangle let's call it Delta 20:30 and the height of our triangle from 20:33 trigonometry we know that the tangent of 20:35 Delta equals the height of our triangle 20:37 divided by its base so 1 over 100 ID 1 20:41 now just like the exponential function 20:43 it turns out that if we Zoom way into a 20:45 tangent function it also looks just like 20:47 a straight line and the slope of the 20:50 tangent function in the neighborhood 20:51 close to zero turns out to just equal 20:53 one so for small values of Delta the 20:56 tangent of Delta just equals Delta 20:59 what this means for our thin triangle is 21:01 that as in approaches Infinity the 21:03 height of our little triangle exactly 21:05 equals our angle Delta when Delta is 21:07 expressed in 21:09 radians now fitting all the pieces 21:11 together here taking some imaginary 21:13 exponent CI Brigg's formula tells us to 21:16 divide C by n this effectively gives us 21:19 a thin triangle with a height of C over 21:22 n and if n is big enough the small angle 21:25 of this triangle is equal to C Over N 21:27 radians now Brigg's formula tells us to 21:30 multiply this triangle by itself in 21:32 times which using the polar form of 21:34 complex numbers means we add our angle 21:37 Delta up in times which gives a final 21:39 angle of C over n * n or just C this is 21:43 why C is equal to our final angle Theta 21:46 expressed in 21:47 radians just as Brigg's formula 21:49 linearized the exponential curve for 21:51 real exponents it's remarkably able to 21:54 work in a similar way for Imaginary 21:56 exponents in the imaginary case 21:59 this happens in the 2D space of the 22:00 complex plane and it works out because 22:03 we can linearize the tangent function 22:05 for very small triangles and because 22:08 multiplying all these little complex 22:09 numbers together results in rotation on 22:11 the complex plane we end up with 22:13 rotation instead of 22:15 growth now that we have oil's formula we 22:17 can finally see how he saved calculus we 22:20 left off with Oiler arguing that both 22:22 livets and beri were wrong and that the 22:25 log of1 has more than one correct answer 22:28 if we take take the natural log of both 22:29 sides of Oiler formula we're left with I 22:32 * thet is equal to the natural log of 22:34 cine theta plus I sin Theta Oiler was 22:38 interested in the case where the 22:39 argument of the logarithm is minus1 this 22:42 will happen whenever the cosine term 22:43 equals minus one and the sign term is 22:46 zero on the complex plane at the point 22:48 minus one Z on our unit circle a Theta 22:51 value of pi radians will work here but 22:54 so we minus Pi 3 Pi minus 3 pi and so on 22:57 so the lthm of minus1 is equal to all 23:00 odd multiples of Pi I this is precisely 23:04 what Oiler means when he says that logs 23:05 have an infinite number of 23:07 solutions oil's Paradox about the 23:10 logarithm of minus1 s now has a 23:12 straightforward solution if we drop the 23:14 two down this means multiplying all the 23:17 solutions of log of -1 by 2 so - 6 Pi I 23:21 -2 Pi I 2 pii and so on now the 23:24 contradiction came up and we squared the 23:26 minus one first instead of dropping it 23:29 down this gives us the log of one using 23:32 oil's formula the log of one will happen 23:34 when the cosine term is equal to one and 23:37 the sign term is equal to zero so at 23:39 angles of 0 2 pi minus 2 pi and so on as 23:43 Oiler points out in his paper the 23:45 solutions to 2 * the log of1 are visibly 23:48 contained in the solutions to the log of 23:52 one this fact along with a number of 23:54 other impressive results convinced Oiler 23:57 that defining log lthms and exponentials 23:59 in this way would save the foundations 24:01 of calculus oil's approach was highly 24:04 controversial at the time but would 24:06 eventually gain acceptance personally I 24:09 don't think it's surprising that oil's 24:11 formula and his approach to logarithms 24:13 was so controversial if we just look at 24:15 Oiler formula it's pretty absurd to the 24:18 right of the equal sign we have cosine 24:20 and sine ways from trigonometry to the 24:22 left we have e raised to a power which 24:24 typically means exponential growth the 24:27 wild card here is of course the 24:29 imaginary unit I which as we've seen is 24:31 almost as if by Magic turning 24:33 exponential growth into periodic wave 24:35 likee motions we began our discussion 24:38 with the idea that many of the equations 24:40 and graphs we spend our time with are 24:42 just Shadows of a more elegant powerful 24:44 and higher dimensional mathematics the 24:47 Deep connection between exponentials on 24:48 one hand and S and cosine on the other 24:51 is perhaps the most stunning example of 24:53 this in mathematics if we take a 24:55 polinomial function like x^2 + 1 expand 24:59 its inputs and outputs to be complex 25:01 numbers and visualize one of these 25:03 imaginary parts of the function as the 25:05 height of a surface above the real plane 25:07 we get these Cool paraboloid Shapes 25:10 incredible but perhaps not a surprising 25:12 extension of parabas to complex 25:15 numbers but when we take the familiar 25:17 exponential growth function of real 25:19 numbers and expand its input and outputs 25:21 to be complex numbers using Oilers 25:23 formula the surface that emerges 25:25 incredibly is shaped exactly like s and 25:29 cosine waves the projection of this 25:31 surface in One Direction is our familiar 25:33 exponential curve and in the other 25:35 direction our coign and sine waves and 25:38 this structure is not some invention it 25:40 literally falls out of our familiar 25:42 rules of algebra carefully applied to 25:44 imaginary numbers all of these pieces of 25:47 mathematics somehow fit together 25:49 perfectly into a form we could have 25:51 never imagined making Oilers formula the 25:54 most beautiful equation in mathematics 26:00 if you want to get deeper into imaginary 26:01 numbers I really think you'll like my 26:03 book way back in 2016 I made a massive 26:06 13-part YouTube series on imaginary 26:08 numbers it's such an incredible topic I 26:11 released an early version of this book 26:13 back then and I'm now in the process of 26:15 revising correcting and significantly 26:17 expanding it my goal is to create the 26:19 best book out there on imaginary numbers 26:22 highquality hardcover printed books will 26:24 start shipping later this year you can 26:26 pre-order a copy today at the link in 26:27 the description below and your order 26:29 includes a free PDF copy of the 2016 26:32 version that you can download today I 26:34 also made a poster version of the table 26:36 from this video that walks through the 26:38 incredible path to Oilers formula this 26:41 is the first dark mode poster I've done 26:42 and I'm super happy with the look and 26:44 feel this poster is a great visual 26:47 explanation of the path we took here to 26:48 get to Oiler formula and I also just 26:51 think it looks cool you'll find all of 26:53 this and more in the Welch lab store Welch Labs 572K subscribers Videos About Patreon 928 Comments rongmaw lin Add a comment... @pendragon7600 1 month ago You pretty much determined the course of my life. You released the imaginary numbers are real series when I was 14 and uninterested in math. That series kicked off my passion for mathematics and I'm now pursuing a PhD in complex analysis and algebraic geometry. My thesis focuses on modular forms. Seeing this video was like a flashback to where it all started. Eternally grateful to you and for the wonderful exposition you produce. I'll buy the book someday (PhD students have no money lol). 1.8K Reply Welch Labs · 63 replies @Kazner0h 1 month ago That final visualization is really stunning. It always amazes me how much a graphical representation can help make a complex subject feel intuitive. 109 Reply @Yotam1703 1 month ago Your complex numbers series was my introduction to higher math education! It’s a pleasure to see you continue it. 659 Reply 3 replies @manuelcastellanos122 1 month ago I'm from Dominican Republic and let's say math isn't much of a thing here. I initially watched your complex numbers series graduating from high school, and I knew a spark started there. Now I'm about to present my thesis for my MSc in Applied Mathematics and it is so rewarding to look back where it all started. Looking forward to get your book! 366 Reply 7 replies @skfineshriber 1 month ago After studying engineering mathematics I gained an enormous respect for mathematicians such as Euler. Here I was, struggling like hell to understand their equations and transformations, while realizing these people had the insight and intellect to INVENT these mathematics. It was awe inspiring. 18 Reply @abpdev 1 month ago I envy younger people who are just getting into math/science because they are starting of surrounded by such amazing free content to explain difficult concepts! I had a hard time understanding this equation in my first and second year. It just seem so odd and counterintuitive. It became one of those truths that I accepted because the textbook said so. The manner in which you presented it here just makes so much sense! The intro, the papers/conversations from the authors, notes, diagram... WELL DONE AND THANK YOU! 120 Reply @sloppycee 1 month ago Euler's motivating example and insight that the i's should cancel; so log(-1) should have an I somewhere is such important knowledge. I hated how they taught math with no context or motivation. This is great! 304 Reply 2 replies @leonhardhoffmann4542 2 weeks ago Addition 2: I really want to give my biggest heartfelt respect and sympathy to you, you do a great job and seriously improved and still improve my life and that of others with your incredibly good videos. I am teaching extra lessons for math for 4 years now (on the side, on high school level) and know first hand how difficult it can be to explain and illustrate mathematical problems and formulas and do so whitout boring or overwhelming the learner, you do an extraordinarily well job here! Thanks! 5 Reply @JohnMartinIT 2 weeks ago The statement that "many of the equations and graphs we spend our time with are just shadows of a more elegant, powerful and higher-dimensional mathematics" reminds me of Plato's analogy of the cave, which I'm sure you appreciate has some interesting implications. Maybe that's why you say imaginary numbers are real; indeed, they may be more real than the more tangible shadows we deal with in this mundane world. 7 Reply @vma011 1 month ago (edited) I discovered your videos on complex numbers 8 years ago, right when I was starting electrical engineering in college. I work with complex numbers and phasors basically daily, all of our electronics, automation mechanisms and electrical grid, work thanks to the properties of these unfairly-named "imaginary" numbers. Sometimes it's easy to get lost on why does this all magic work. So glad to be able to come back to your channel and find this ever growing content in the topic, now materialized in your own book. Congratulations for achieving that huge milestone, I will be definitely looking into getting a copy of my own! 4 Reply @Mytubecleaner 1 month ago 15:00 wow, that was an enlightement moment for me 34 Reply 1 reply @rishabhjohri4223 1 month ago I was 15 years old when I was introduced to this channel, which fuelled my interest in mathematics as well as artificial intelligence ( the "learning to see" Playlist on decision trees), now I'm 23 years old and pursuing Master's in artificial intelligence. Thanks for this amazing and free content! 5 Reply @BooLightning 1 month ago Euler was like, nahh bro you’re both wrong 1.3K Reply 12 replies @ericdculver 1 month ago If I had the time to truly teach Euler's formula instead of just mention it in passing, this is exactly how I would want to teach it. Great job! 18 Reply @thulioassis1023 1 month ago This reminded me of my teachers in high school. All brilliant teachers who explained the reasoning behind the formulas. Math can drive me crazy because I can easily become obsessed about a problem or a solution. Thank you for the video. It really sparked something inside of me. 4 Reply @fyu1945 1 month ago 9:33 You define a conjugate as "flipping the sign of the imaginary component" but what you're doing here is not flipping the sign of the imaginary component, but flipping the sign of the exponent, which happens to be imaginary. Of course in reality it amounts to the same thing, but that's what we're trying to prove! We don't know yet if 2^(-bi) is the conjugate, so we can't use this fact to prove the magnitude of our number is 1. 147 Reply 14 replies @jaewok5G 1 month ago i think this is the first video of yours i've ever seen. i understood every bit of this but only ever as separate bits. bridging the gaps of what was once left as "details that we won't go over in this class" gave me the small thrills of recognizing, "hey, that's practically e" and "hey, that's really close to a radian." fun stuff. 3 Reply @galileo3431 1 month ago Gosh, your flow of delivering information is so smooth! Subbed! 2 Reply @vari1535 1 month ago (edited) 14:42 holy crap, i FINALLY understand the actual MEANING of this definition of e. thank you so much!!! (the video as a whole (and the poster summary, holy crap!!) was also AMAZING.) 4 Reply @ryrylandcripps5811 1 month ago e^iPi=-1, my favorite. An exponential raised to the power of an imaginary number multiplied by an irrational equals an integer? Beautiful. 2 Reply @kiko7247 1 month ago I just love these videos. The topic, animation, it's entertaining and educational at the same time. I love complex numbers and the history of math. Thank you. Your efforts are greatly appreciated 27 Reply @pauls5745 1 month ago Why am I seeing aero/hydrodynamic shapes in the graphs of these? Too elegant and simple! There is deep beauty in math. 2 Reply @iveharzing 1 month ago This video is a beautiful explanation of Euler's Formula that I haven't seen before anywhere else. I subscribed to you after watching your first imaginary numbers video, and I keep being impressed by what you upload! 32 Reply Welch Labs · 2 replies @bender5835 1 month ago dude, I binged through your complex number series few years ago and still to this day I share that playlist with people. So looking forward to the book. Amazing visualizations btw. what a time to be be alive man. Ah, I am getting chills from excitement. Ty! Ty! 1 Reply @caio.tavares11 1 month ago Man that book looks amazing! Please consider adding international shipping :) 10 Reply 1 reply @ArshiaSa-ku2qd 2 weeks ago I saw your masterpiece of a video on imaginary numbers 5 years ago and i still think about it some days. You were the only one that could explain the topic in a way that elevated my understanding of the topic. Reply @akshatrai9007 1 month ago This channel is criminally underrated 121 Reply 2 replies @gdibble 1 month ago ᛣ Well done and [well] said. The elegance of this video and your style earned my Subscription. Just want to emphasize my appreciation for the time you took to produce and release this video. Keep up the excellent work in maths and inspiring learners everywhere! 👏 1 Reply @edwarding4355 1 month ago If I learned math in a historical way, I would have understood it better 3 Reply @TazariaGaming 1 month ago The way that fully expanded form of the formula comes together as a combination of trig and exp functions is just a beautiful conclusion to this story. Your videos on imaginary numbers was what sparked my interest in them way back then. Thank you for making another video on the topic! 2 Reply @itscoolto1 1 month ago Nobody tell Terence Howard about this <.> 23 Reply 2 replies @billynomates920 1 month ago that was one of the best maths youtubes i have ever seen. 1 Reply @luisclovis09 1 month ago Thanks Euler for discovering the mathematical base for eletrical engineering. And thank you for making me able to visualize it 😄 11 Reply 1 reply @prabodhshejwalkar 1 month ago Your complex numbers series was my introduction to higher math education! This changed my life! Reply @cem_kaya 1 month ago i was playing around with numbers at a math class in high school. i accidently put i over 7. i could not understand how such a number would work on my own. i asked where is this number on a imaginary number plane to my math teachers. Some answered that is illegal, some admitted they did not know. One of my teachers (Ömer hoca) tried to explain it using the Cis function and a formulaic approach. To be honest, until that point i could derive the intuition behind most of the math and physics stuff on my own. This problem where is the number 7^i was impossible for me and no one around me cared about the intuition of things. One day YT recommended imaginary numbers are real to me. That video made me look at the internet from a new perspective. i learned that it can be a great tool for education and self learning. i thank you for widening my horizon. Evresince, the internet has been the most valuable resource in my life. Now i am studying ai at a masters level, thanks. 11 Reply @tenma628 3 weeks ago (edited) such a great video !! i was astounded when you revealed that the exponent function's imaginary surface looks like a sin wave as it looked almost exactly like how i imagined it before you showed it. such a cool relavation !! Reply @aglargalad 1 month ago Never appreciated that Euler's equation visualizes 2D exponents, sin, and cosine functions in 3D! This was an awesome video! Thank you! 20 Reply 1 reply @stevenverrall4527 1 month ago (edited) Euler's formula is the most beautiful and useful stand-alone equation containing trigonometric functions. In my humble opinion, the most beautiful pair of equations involving trigonometric functions are Lynch's equations. They were likely known classically, but are almost unknown to modern mathematicians. Lynch's equations are shown in the Wikipedia article Lemon (geometry). 1 Reply @TheSereneWanderer87 1 month ago After watching this beautiful video, I realize that I have studied nothing in school. 6 Reply @PASHKULI 1 month ago Great update on the series! I watched it back then and was so grateful someone (you) had taken the time to explain and visualise and elaborate on this amazing subject! Reply @kushagra64 1 month ago This was the first time I felt e^i π = -1 is beautiful result... 32 Reply 5 replies @girishrajgor9791 1 month ago You are Flashlight on Path of Math....No words to explain for your hard work...carry on.. Reply @blueslime5855 1 month ago At 9:10, the angle in blue should be arctan(2)≈63.4° You may have accidentally calculated arctan(1/2) 14 Reply 1 reply @NetTubeUser 1 month ago Your video is exceptionally slick, clean, professional, and impressive... and informative, indeed! You did an outstanding job. Congratulations. Reply @yash1152 1 month ago 4:03 okay whits. stay bound. 3 Reply @Sagitarria 1 month ago I want to say two things one your work is phenomenal. And you are maybe the only YouTube creator that I have ever given constructive criticism to who has respectfully taken it. I have so much respect for you and your work Reply @erawanpencil 1 month ago This is the best video explaining logarithms of negative numbers I've ever seen... thank you for walking us through the steps, it helps for beginners like myself. Explaining our ordinary real number arithmetic as "shadows" of a truer reality in the complex space was evocative.... maybe you can make a video on the Riemann sphere some day or recommend some others. When Riemann put a light source at infinity and showed that Möbius transformations are just shadows or projections through that sphere, it seemed like it was telling us something about reality way beyond just complex analysis. 6 Reply @DFeltran 1 month ago (edited) What a great video! Congratulations for the high quality content and great explanation! Reply @fluffpuckot 1 month ago At 9:15min: in my world 18.4+26.6 = 45, not 55. And to think they taught us not to "drink and derive", but it is obviously good enough to spot a basic error. 4 Reply 2 replies @johnraffensperger 1 month ago Perhaps one of the best videos I've ever seen. Very well done. Thank you 1 Reply @supermuffinbros4797 1 month ago 9:33 It's not clear why this makes the complex conjugate at this point in the explanation... 23 Reply 1 reply @Saki630 1 month ago your 2016 video series on Imaginary Numbers was critical to my understanding the subject during my advanced mathematics courses for my M.S. ME. I never forget the hard work you did showing the mapping of the complex plane. Reply @mrolemartinruud 1 month ago Any plans on providing international shipping (Norway)? This series was a huge part of my understanding of the complex numbers, and I would love to support you through buying the book. Thank you for the amazing videos. 6 Reply 1 reply @AidanDaGreat 6 hours ago Euler was friggin awesome, man. Bro lost went almost totally blind in his left eye, then totally blind in his right eye. And my bro really said, "Now I will have fewer distractions," and made stuff like this for the rest of his life. Total homie. Reply @angeldude101 1 month ago We can go deeper. Complex numbers actually have 2 much lesser known cousins (lesser known because they don't actually expand what's algebraically possible). If we can define a number i such that i² = -1, why can't we define other non-Real numbers with Real squares? From this idea, we get the hyperbolic numbers (usually called the worse name of "split-complex numbers), and the dual numbers, defined respectively with j² = 1 and ε² = 0, though with j and ε not being Real despite their defining equations having Real solutions. Now what happens when we take their exponents? e^j and e^ε? Can we do the exact same thing we did with imaginary numbers? Of course we can! They even display similar "angle" adding behavior, though with different notions of "angles". I don't really see much magic in Euler's formula anymore... because the magic is entirely within the exponential alone, and Euler's formula is merely one manifestation of the beautiful exponential function. exp(φi) = cos(φ) + isin(φ), exp(φj) = cosh(φ) + jsinh(φ), exp φε = 1 + φε (wait a minute... that last one looks suspiciously similar to the the limit definition, but when N is set to 1. In addition, while I initially said that j and ε aren't Real numbers, the exponential does not care and the three formulas I gave work regardless of what i, j, and ε are as long as they satisfy i² = -1, j² = 1, and ε² = 0. Even j = -1 works in the hyperbolic case.) Also do I need to mention it? Because Euler's formula works completely unchanged for quaternions. Just normalize and pretend it's i. The exponential might be my favourite function, and it bothers me how many sources try to explain it with the Taylor series, rather than the beautiful geometry of change proportional to the value, and the algebra of the bridge between the multiplicative and additive groups. 11 Reply 2 replies @tracegate9759 1 month ago I've been a math tutor for my old community college the last 4 years. I've lost count of the number of times I've seen student's eyes change as the light bulb goes off in their head because I can explain where many seemingly "imaginary" concepts actually come from (ha). It's all because of well thought-out explanations like this that I can propagate while assisting with homework. Thank you so much for your passion, it truly has an impact! :) Reply @sb-sm8ib 1 month ago (edited) This is wonderful, I loved it ❤. I always had this question about what it means to raise number to imaginary power. Besides i was hoping if you included interpretation of irrational powers. 5 Reply 3 replies @kaabirali1 1 month ago I love this explanation. Fundamental questions are truly fascinating. Reply @tedsheridan8725 1 month ago Even cooler - you can plot e^(x+yi) in 4D space, and see a continuous manifold that captures the entire complex function. 6 Reply 5 replies @QP9237 1 month ago I always enjoyed your complex numbers videos. I saw them right after I took complex variables in undergrad, and got to say it's what kept pushing me forward to reading and collecting complex analysis textbooks (single and several variables), and I love it so much. I even am starting to apply some of the logic into my medical career for research/modeling purposes. Also my all time favorite number is Gelfond's constant, because it's the power relation of the negative real and imaginary units producing a positive transcendental number! Reply @abdulmoeed4751 1 month ago At 9:47 how can we assume that conjugate of 2^bi is 2^(-bi), in general conjugate of f(z) is not f(conj(z)). P.s love your videos tho ❤ 22 Reply 2 replies @MaeleneReynolds 1 month ago I chose to help and share the video in the social media. I hope the popularity will grow. Reply @sss-chan 1 month ago At 9:33 why complex conjugate of 2^ bi would have a form 2^-bi? I mean it's clear from Euler's formula, but how to get it without proving it 43 Reply 10 replies @Vannishn 1 month ago 9:33 It is really the conjugate you multiply ? I think it's the inverse ! 2^1 × 2^(-1) is one but the magnitude of 2 is 2 7 Reply 1 reply @gregvaughntx 1 month ago This is an instant subscribe for me. A few years ago I was talking with my teenage kids about tattoos. I said I can't imagine anything I would still think is profound 10 years later ... except maybe my favorite math equation, Euler's Formula. Their jaws dropped and they called me such a nerd for even having a favorite formula. Reply @Cpt.Zenobia 1 month ago If the graph you drew @3:39 is of log(x), then the slope at x=-1 is not 1. looks like d/dx at -1 is -1 and at 1 is 1. does not make sense! looks more like graph of 1/-x^2 + 1. 5 Reply 1 reply @aglawe1 1 month ago I was taught this in high school but never was able to understand how exponentials had to do anything with trigonometry. The way you explained in the video both historically as well as mathematically blew my mind. Surely it is hard to accept this equation without being able to visualise it. Thank you so much for your effort and keep up the good work 👍 Reply @spitsmuis4772 1 month ago 9:35 Wait, how do you know that 2^(-bi) is the complex conjugate of 2^(bi) ? 6 Reply 1 reply @InterestsInEverything 1 month ago I have followed this channel since its inception, and it has been wonderful! But I want to point out one small thing: You said (at 2:42) that we broke away from the "multiply by itself" definition. This is actually a misconception based on what people think "negative" means. Since "negative" simply refers to "opposite" (negatives are the numbers opposite the positive on the number line), then a negative EXPONENT just means a negative operation (the opposite operation of multiplication is division). Exponents raise the POWER, and operations are an increased power of a stagnant object. Thanks for all your hard work, Stephen! 1 Reply @kristoferkrus 1 month ago 9:28 Why do you say that 2^(-bi) is the complex conjugate of 2^(bi)? 6 Reply 4 replies @tyagiabhi 1 month ago Beautifully explained! Thank you for this video. Reply @angrydachshund 1 month ago @1:01 Congrats on your book. From the subject matter, does that make you a fiction author? 5 Reply @CherryNerd27 1 month ago Thank you so much for making this beautiful video! You made all the different pieces of maths fit together so effortlessly. I hope that I never forget this explanation in my life. Reply @fullfungo 1 month ago 9:30 unfortunately, you are wrong here. You previously said that if (a+bi)•(a-bi) = c Then |a+bi| = sqrt(c). But now you are saying that 2^bi and 2^-bi are conjugates without any explanation, except that you replaced bi with -bi. This does not work in general. Consider f(x) = 2i•x. Then f(bi) is -2b and f(-bi) is 2b, but they are not conjugates. You got lucky here. If 2^x was equal to, for example, 1+2i•x-x^2 (or some other expression) Then 2^bi and 2^-bi would not have been conjugates. You need to prove that 2^bi is decomposable into f(b)+i•g(b) where f is an even function and g is an odd function first. 7 Reply @anastasijajamrik2978 3 days ago This is one of the best explanation of raising numbers to the power of a complex number and how e come up in this concept. Just so easy to understand!!! I had to rewatch this again because its so nicely explained! Thank you so much, this really helped me understand this weird concept!!! Reply @stefanogiorgetti417 3 weeks ago This video feels like a complex magic trick in “the prestige” movie - except that is mathematics!! Truly mind blowing!! Reply @TECHN01200 1 month ago Your complex numbers series was how I came to learn and understand complex numbers. It is one of only a few, if not, the only one that really exists. It was phenomenally done. Unfortunately, I didn't receive any education on them in school, so I must be greatful to you for teaching me the math I would've never been able to find elsewhere put so well and intuitively. The image of you pulling a parabola out of the page was such a great visual. 2 Reply @Cuerdoylisto 1 month ago Gracias por compartir su maravilloso trabajo. . . Reply @PlexiumGames 1 month ago Euler is the greatest of all time and nobody can convince me otherwise. Reply @carolinalp 1 month ago Amazing video!!! So clear, so beautiful as Euler's equation itself. Thank you! Reply @nikwakem 1 month ago (edited) Awesome quantity of information taught clearly in 20mins Reply @SreenathSreekrishna 1 month ago After working with complex numbers for over a year, even using euler's identity, I never really understood it completely until this video. Thank you for a fascinating explanation! Reply @MrCoreyTexas 1 month ago (edited) haven't gotten the chance to watch your video yet, but your thumbnail in the list of related videos really stood out visually to me! Reply @TheJara123 1 month ago So, the man comes to his senses and realised he has great gift to explain complex analysis and we the people are in need of it... So he leaves his other topics and presents us this beginning. With great gift comes great power And with great power comes great responsibility.... present us the entire complex analaysis...all the way to advanced complex analysis and you got your true followers... Man, a million thanks on behalf of L.Euler. Reply @trevorgrover5619 1 month ago (edited) Complex numbers, euler's formula, Laplace transforms, etc really simplify electrical signal and circuit analysis. Reply @underfilho 1 month ago your playlist in complex numbers was my insight that I really love this, and now I'm doing my undergrad thesis in Complex Analysis, thanks. I'm doing it by connecting Complex Analysis and Harmonic Funtions, using it to solve the Dirichlet Problem by the Riemann Mapping Theorem. Reply @GoldenAgeMath 1 month ago Preordered the book immediately! The original imaginary numbers are real series is one of my favorite pieces of math exposition and one of my favorite things on YouTube of all time! I hope to someday create something half as good! Reply @johnferrara2207 1 month ago The hardcover is at the top of my Christmas list. REALLY looking forward to this. 1 Reply @siddharth-gandhi 1 month ago I’ve a playlist called great videos where i save videos which really resonate. So far i think all your videos I’ve watched have gone to that playlist. Stellar job good sir! Hoping you keep at this for many more topics Reply @shurturgal 1 month ago Amazing video, I really liked the ones you made some time ago, and even made a playlist out of it so that I can watch them easily whenever I want too. Thank you for your work! Reply @evasuser 1 month ago We are lucky that videos like Welch (and 3b1b and others) exist and fortunately yt still allows them to exist. 1 Reply @hichams4445 1 month ago People like you make me love the internet and math. thanks a lot Reply @davidbarts6144 1 month ago I was SO frustrated when I learned about imaginary numbers in high school and was then told that my hunch that logarithms of negative numbers must be imaginary was nonsense. Then I felt SO vindicated when I learned about Euler’s formula in college. 1 Reply @xfry 1 month ago OMG I love your Imaginary Numbers are real and I go to buy it. Thank you for promoting love for math with your videos! Reply @wacharaboy 1 month ago No wonder you have so many compliments in the comments. You are really GOOD explaining. Never thought I could be even able to grasp some of the knowledge you provided. I've subscribed and I haven't even finished the video: with experience now I know when a teacher is actually able to make me learn something... Reply @phrozenwun 1 month ago (edited) Even though you touch on differentials, I think you omit one of the more elegant demonstrations; the idea that e^x is its own derivative allows you to move to an expanded representation of e^x, cos and sin demonstrating equivalence. Beautiful presentation none the less, thank you for sharing. Reply @kevinboles3885 1 month ago kudos on the book! you picked a fascinating topic for sure Reply @belg4mit 1 month ago This is a really good explanation of why these things are connected, so often one only is told/roughly shown that they are. Reply @edwardlulofs444 1 month ago I used the author Churchill for complex analysis. It impressed me then and now as a collection or framework of incredible simplicity and tremendous power. Now when someone says “complex analysis “ I think Cartan algebra. Very beautiful. Thank you for a wonderful video. If I taught complex numbers I would start by showing your video. Reply @danielduckq 1 month ago this is probably the most favorite math video i've watched Reply @ferriswhitehouse1476 1 month ago I took some courses on complex numbers for electrical engineering 8ish years ago and this was such an awesome refresher for me. I remember thinking that complex number math was one of the greatest achievements of humanity. This video is an absolute masterpiece, I followed perfectly start to finish through a whole bunch of stuff that I thought I forgot. The context, the timeline, the visualizations, the narration, everything is just so awesome. Thank you! 2 Reply 3 replies @XxMrRoachxX 1 month ago (edited) You gained anew sub!!! Now please make a video and turn those 2D representations into actual 3D (maybe 4D) waves! :) Like actual E and B fields. Maxwell equations and such! 1 Reply @ryanchristiansen 1 month ago Mind blown at 25:33! The nature of mathematics is so perfect. Reply @luigiwilkins8806 1 month ago (edited) Awesome vid. I learned of Euler's formula in my circuit analysis class, but it was not really explained, so I find this video really insightful. I love your videos, and I always leave with something new. Reply @markproulx1472 1 month ago I smiled during this entire video. 1 Reply @valentinfelsner277 1 month ago This video nicely sums up one of the biggest achievements of "the internet": access to world-class teaching material in our home. And it shows that unfortunately, so many people had given up on maths early in their live most likely due to lack of teaching skill available to them at their time. Reply @NoiseWithRules 1 month ago Ever heard of Napier? He did the hard work of calculating logarithms for fractional numbers. He then collaborated with Briggs who then recast Napier's work to base10. 1 Reply @espartacodepaola 1 month ago Wonderful explanation! Relates the historical or more precisely the human relations and the discovery behind it. Showing that science is much more than cold numbers and results. Me as an engineer with 60 yo have goosebumps seeing it. Reply @alaricgoldkuhl155 1 month ago Euler's Identity is the most beautiful phrase in mathematics. Especially when you consider that God is One. 1 Reply @dewinmoonl 1 month ago (edited) 15:20 as a computer scientist I really appreciated this guy's conviction. he developed an elegant algorithm ahead of computers, which would've been SO tedious to do by hand. however, he knew what he had could not be appreciated by the lay-person w/o something practical, so he went with the hard work and computed a table by hand. the table is likely to be hugely, practically useful. through this practical table of values, his brilliant algorithm, along with his mathsmatical "side product" of that analysis survived until now Reply @mikejones-vd3fg 1 month ago (edited) Love the historical context of mathematics, makes it feel more alive, people working on problems going back and forth with competing solutions, here we've jsut accepted everything, i dont see anyone with their own ideas. Part of our education system didnt teach us to think for ourselves and explore on our own, we were taught to learn othe peoples work and pass tests. Reply @Misugarwwx 1 month ago (edited) Damn this video singlehandedly explained where the formulas for the value of e derives from, imaginary exponents, and euler's formula. I really wanted to know the history of this unexpected but incredible constant, but the amount of text was daunting so I couldn't ever start. This video's amazing man thanks for brightening my day :) + also pog visuals Reply @thejuanderful 1 month ago That was so beautiful I was in tears at the end! 💜 Reply @4m0d 1 month ago I don't yet fully understand the series but I am grateful for your effort, and it is so cool you wrote a book so we can get an intuition behind the beautiful complex numbers. Reply @andrashorvath2411 1 month ago Amazingly great job at explaining it at the right pace with the right info and nice animations, cheers. Reply @mikeblake9761 1 month ago Love being a normal person who’s just fascinated in maths and physics being blown away by how awesome humans are sometimes Reply @iccuwarn1781 1 month ago Fantastic video as always! You're really setting the bar high for all the other math youtubers :) Reply @toughenupfluffy7294 1 month ago When I started watching this video, I never imagined I would be able to follow it, but somehow I did. Well done! Reply @mb10mb10 1 month ago What a wonderful lesson. Thanks! Reply @physics_enthusiast_Soorya 1 month ago Omgggggg!! I'm really exited to watch this video!! Feels like it's gonna blow up I really wanted to understand colours in complex numbers they show in Quantum physics from quite a long time. Hope my brain will help me understand it through this video.. 😂😅 I'm Grateful Youtube, Thanks 🫶️ 1 Reply @mayankraj2763 1 month ago finally understood what e^(it) = cos(t) + i sint(t) really means. Can truly appreciate how it was a gemstone hiding in the mathverse and how it was clinically hunted by euler. Reply @kellymoses8566 1 month ago Getting the formula tattooed on your arm at the end of the video is quite the mic drop. 1 Reply @exp9r 1 month ago This video deserves a Pulitzer! Reply @skpcboy 1 month ago I remember learning imaginary numbers with this channel as a middle schooler. Oh how time flies. Reply @TheHilcros 1 month ago Eye opening. Thank you! Reply @EconAtheist 1 month ago Geez I've seen all sorts of Euler's Formula breakdowns on YouTube -- this one is the most fleshed out, me likey! New subscriber here. /also, early in the video, you immediately started rotating your 3d extrusion so i didn't have to burn brainpower thinking of how it should look //yes! Reply @Tom-gd4pb 1 month ago with this video everything falls into places for my understanding Reply @primenumberbuster404 1 month ago This channel is the best thing on youtube right now. 2 Reply 1 reply @audbee14 12 days ago I’m a mere pre-cal student, but when I learned about imaginary numbers in algebra II, I was super interested. I don’t yet deeply understand everything you said in the video as I haven’t learned that much calculus, but it certainly verifies my interest of complex numbers. Thanks for the amazing video, and educating me on things I can’t wait to learn! Reply Welch Labs · 1 reply @nimzovidal 1 month ago I'm glad I found your video. I'm a visual person, and while I've always enjoyed math, I had difficulties "seeing" math. While most if not all of what you've put together in your video I have probably heard it at one time or another, Today, your video unlocked the possibility to go deeper in my engineering/physics understanding. Caveat - I've never had to memorize my trig identities because I just learned how to derive (most of) them from Eulers equation (knowing the history just made it better). I went ahead and purchased your upcoming book, and subscribed to your channel - please continue your work. How about something on Greens functions? Reply @popcorn485 1 month ago (edited) Your original video was a huge hit! This is a fantastic follow-up! Absolutely excellent! (Edit: And heaps of great content in between I should add.) Reply Welch Labs · 1 reply @octobermathematics 1 month ago Thankyou. Wonderful Video 🙏 Reply @ankurantil6137 1 month ago This is probably the best video about Euler's equation that I have seen Reply @SuperLlama88888 1 month ago I thought it would get really complicated after the first 5 minutes, but you explained this really well! Thank you! Reply @hish_mat 1 month ago This video is just awesome! Thank you! Reply @ToolTechSoftware 1 month ago What a wonderful film!! Reply @jerielczy 1 month ago I remember watching this in secondary school damnn. I remember binging your videos at home Reply @FullCircleTravis 1 month ago What I would love to see in a math video, is application. Theories are fun, but most people gain way more interest in learning when it has a application. For example, I had to build a PID fan controller for temperature regulation by exhausting heat. To do this, I had to learn calculus. I had never bothered with calculus until it was needed for an actual application. Reply @k.nitishjoshi1875 1 month ago i sometimes feel that how this type extremely intellegent type of people were and are among us. Amazing 1 Reply @akademesanctuary1361 1 month ago Only a true love for imaginary numbers could produce such a beautiful presentation. Well done! I'm in the editing stage of an imaginary numbers book that goes into how they logically work, form and transform in nature. It shows how two levels of imaginary numbers can be constructed using dual numbers with one or two axes (tessarine like triplexes and quaternions), how they relate to potential, the line element, evolve to octonions and the metric to show how spacetime is assembled to form various energy states like matter. Reply @thingthingthingthingthingthing 1 month ago I love math because it’s infinitely (almost) big and I always have something to learn and discover Reply @threeuniquefingers 1 month ago Waitttt your'e the guy who had the imaginary numbers are real series!!! Ahhh SO glad that I REDISCOVERED YOUUU. I watched that when i was in grade 8!!! ANd now i'm in bachelors 2nd semister...wowww seems sooo nostalgic Reply @gtagamerm4169 1 month ago was waiting since i've seen your contents on Reimann surface 1 Reply @bobthecomputerguy 1 month ago I really learned imaginary numbers are real when studying electronic circuitry and the math that models their behavior. That was 25 years ago, but this is the first time I've seen the 3D explanation that shows exponential growth on one plane, and sinusoidal behavior on another. That was both mind blowing and "yeah, that makes sense" at the same time. Reply @dieuwer5370 7 days ago (edited) Using the trick of stacking triangles, you could calculate Pi, considering that Pi is nothing more than stacking triangles until you are halfway the circumference of the unit circle. Therefore, : π = (180/δ)*tan(δ) for δ << 1. Another fun thing coming out of Euler's Identity is: ln(i) = 0.5πi. Meaning, π and i are related. Reply @michaelpeeler7030 1 month ago This was an incredible video. Thank you. Reply @bmavad 1 month ago Keen to see you have a bash at a complex version of Euler's Gamma function. Reply @brockobama257 1 month ago No way a fucking SEQUEL to my favorite video series of all time that I saw a decade ago? When i was in high school? That’s insane thank you welch, btw im the guy who asked you on tiktok for the end credit song, the pretty bells and piano keys made me happy thank you Reply Welch Labs · 1 reply @_zh3ro_ 1 month ago I have no words. This is how math should be taught. Reply @Josef-ed7dk 1 month ago bro you straight up explained the relation between exponents and logarithms better than my high school teacher. bravo keep it up Reply @ericm301 1 month ago That graphic at the beginning just makes my brain tingle! Reply @josephhargrove4319 1 month ago A most wonderful video! You explain a very complicated, abstruse topic in probably as simple a way as can be done. You managed to increase my respect for Euler, who I already considered the most significant mathematician this side of Archimedes. And you didn't even mention the special case of Euler's formula, Euler's identity, which many, many mathematicians consider the most elegant equation in Mathematics. richard -- The sentence "i is complex." is syntactically correct. Reply @smartvonooo4823 1 month ago (edited) I thought this video was made 5 years ago considering how good it is, i would’ve already watched it Reply @TheScienceNerd100 3 weeks ago 19:30 This part hits so hard after going through college and using that formula, never knowing where it came from. Now that I know how it came to be, it all makes sense. I knew that angle of 57.296 sounded so familiar Reply @GreaTeacheRopke97 1 month ago Strong opening, reusing that parabola. I show a clip of that video in my classes every year. Absolutely blows minds away. Reply Welch Labs · 2 replies @repeatatron1 1 month ago Can't wait for the book. Reply @qu4rc 1 month ago I'm 16 and have had an interest in complex mathematics for a while, but this is incredibly interesting, albeit 50 percent of this flew right over my head. Reply @omargoodman2999 13 days ago Reminds me of the model where shining a light on a tapered cylinder will produce either a circular, square, or triangular shadow on the wall depending on the relative orientation of the object to the light. Trying to define properties of the shape based on a fixed view of one shadow will lead to very incorrect conclusions no matter which angle it's from. Euler examined it illuminated from multiple directions to work out an overall shape that was impossible to physically "see", implied by shadows cast at different "angles", mathematically speaking. Reply @denelson83 1 month ago The argument of 1 + 2i is actually about 63.4°, the complement of the argument that you specified for that value. 2 Reply @Paul-ty1bv 1 month ago Seeing radians fall out from Briggs' equation into Euler's was revelatory. It was beautifully explained visually. I think I understand it much better now. Still don't understand it all, but thank you for this. Liked, subscribed, and recommending. Reply Welch Labs · 1 reply @ugestacoolie5998 2 days ago lovely video, it's like a walk in history of how this piece of math came to be, absolutely loved it over some long formal proof, or smashed in my face by some professor to memorize it for some test. Reply @tariqpahmed 1 month ago e explainer was dope ... thanks for that!!! Reply @Enzo_1098 1 month ago thank you so much for all these vids! it really made my childhood special Reply @atimholt 1 month ago One way to think of it is that euler's formula reframes what you're actually doing when you exponentiate. Repeated continuous rotation vs repeated continuous multiplication? They're both linear transforms. So it's really about continuously compounded linear change. Reply @diraziz396 1 month ago Wonderful. little by little i'm getting there Reply @phillewis3108 1 month ago I feel like my mind should be blown, but I have no idea what just happened. 1 Reply @PitchWheel 1 month ago Superb video 2 Reply @macronencer 1 month ago There are sqrt(2) kinds of people: real and imaginary. You're for real. Thank you for your hard work! The book looks great. Reply @yizhang7027 1 month ago Very nice visualization at the end. Reply @nosuchanimal6947 1 month ago i feel like, if i watched this a couple more times, i could wrap my head around that enough to have a much better appreciation of animation vs maths Reply @TiagoCavalcanti-ji6hu 1 month ago Those 27 intense minutes, bruh! It is the most beautiful thing I've seen. Reply @PeerAdder 1 month ago Brilliantly done. Reply @megumiarc 1 month ago Mathematics is really beautiful. Though i usually curse out the formulas for making no sense and being complex, discovering things like these reminds me how they come into existence and how beautiful it is that something as constant as maths stays in this universe. Not everything here is constant, such as the concept of quantum superposition suggests, and that makes math ever so fascinating to me. Loved this, I'm glad i went back 3 times when i started to get confused instead of leaving the video in between haha. Reply @CssGamer28 1 month ago I really love the front page of your book! 😊 Reply @paulkolodner2445 1 month ago The most amazing thing about this beautiful video is that it has gotten almost a half a milion views. Reply @enriquemacchiavelli8771 1 month ago Fantastic!!!! Thank you Reply @AlejandroSanchez-ib6hj 1 month ago (edited) How did you know I needed a book about that series? Ordering ASAP.... just realized it's only delivering to USA. Please add Mexico! Reply @tonygale5455 1 month ago Should have done the tattoo of the formula on the forehead instead on the forearm. Amazing explanation into visualisation. Fantastic work. Wish I had this explanation at high school visually. The internet and computers didn't exist back in my high school days. Reply @bgold2007 1 month ago (edited) the 3d paraboloid 24:42 and imaginary exponential 1 Reply @Schopenhauer667 1 month ago Im less than 1 minute on the video and stopped to comment that the editing of this video is simply amazing. Reply @edilmolinafernandez7670 1 month ago Excellent video, my friend, after years of studying mathematics and dealing with exponentials and logarithms of complex numbers, I finally have a solid intuition of what it means. Thanks!!! Reply Welch Labs · 1 reply @KeithRowley418 1 month ago What a beautiful explanation! Reply @federicodgossetti 1 month ago I do like it on the form: e^(iπ) + 1 = 0, containing the constants e, π, and the values +1, 0 and i. Reply @michaelhughes6634 1 month ago And then this incredible formula leads to a more deeper understanding of reality with the famous Schrödinger equation. It can not be overstated how important eulers equation is for Mathematics and Physics. Reply @thenixaless7493 1 month ago No way you tattooed Euler's formula on your arm you legend!! 1 Reply @cjwilly1862 1 month ago (edited) The Taylor series expansion of e^z around z=0 (Maclaurin series) makes a pretty strong argument for laying bare Euler's formula. Reply @ErikS- 2 weeks ago (edited) Compliments for this amazing animation: 24:38 Reply @jonphelan707 1 month ago (edited) In a world where i takes its stand, Euler's formula links it to the land. With e to the i pi, so divine, It shows - 1 in a grand design, Making math's mysteries perfectly planned. Reply @thisthatnthethird123 4 weeks ago Euler, you the man! Reply @brygos7436 1 month ago Thank you! Reply @hubertorhant8884 1 month ago Just plain awesomeness !!! Reply @AlexCFaulkner 1 month ago What ever that first thumbnail was I passed over this...but new one was way intriguing Reply @tsingtak642 1 month ago I am excited that there is a new video about complex number released by Welch Labs Reply Welch Labs · 1 reply @briandonovan9091 1 month ago That's was my vote too, for a Quora question about the best equation. It is amazing, I actually used it in audio processing stuff. Reply 1 reply @ruperterskin2117 1 month ago Cool. Thanks for sharing. Reply @edercuellar2694 1 month ago This is why I love history of mathematics. Reply @doctorwilly 1 month ago when i saw the formula for the first time, i was like how the hell did anyone to come up with this? How does this make any intuitive sense?? thanks to you I can finally understand today. Reply @roy-ub7sy 1 month ago Amazing explanation Reply 1 month ago Quite satisfying. Thank you. Reply @gvd-l3o 1 month ago Amazing video, great job! Reply @05degrees 1 month ago After already learning this stuff to some degree I find these historic scenes very delightful! Despite one should’t try to repeat history to learn math in the first place but better use a streamlined, simplified ways, learning what happened later is… somehow even just aesthetically pleasing. And sometimes we probably forget too much and lose some gems. (Well, because one can’t just be lost in gems while learning basics, either!) I’m glad to have liberty to be entertained by this stuff, thanks for doing this for us! ⚗ Reply Welch Labs · 1 reply @rebellischercherub849 1 month ago Beautiful 1 Reply @markcrites7060 1 month ago Watching you develop this using degrees for your angles instead of radians was excruciating. It's so much easier using phasor notation. Reply @Strifeart. 1 month ago This is what Terrence was trying to conceptualize with Eric. Sacred Geometry is enlightening to say the least. Reply @chisomokalumbe445 2 weeks ago I finally found someone who gets it too. We need to start computing in 3D always. Reply @musa_vnscnt 1 month ago I'm buying that book🚀 Reply @davidbrozovic 1 month ago (edited) 9:15 best voice crack ever Reply @freelancing_101 3 days ago Thank you fellow Welch Reply @renezirkel 1 month ago 25:35 Nice graphic. I have never seen this before and it helped me to understand better, what is going on. Thanks. Reply @jursamaj 2 days ago 9:30 At this point in the argument, it's only an assumption that 2^bi and 2^-bi are complex conjugates. They have to have opposite polar angles, and their magnitudes have to multiply to 1, but it could have been that one was outside the circle of radius 1 and the other was inside (just as 2^b and 2^-b are). Reply @jimburd58 4 weeks ago Minor correction: at the 9:17, you show that the sum of the two angles (18.4, 26.6) is 55, when it should show 45. Reply @MsXXmetalheadXx 1 month ago I don't like complex analysis but your video made me change that view, i'm going to give it another try. Reply @yopenzo 2 days ago Very very nice work dude! Reply @mattanova 1 month ago I had an A-ha! moment in this video comparing this to the wave equation. It makes wonderful sense why the equation has the exponential part for its propagation. Also, I like how you present things, please become a Professor (if you aren't one already). Reply @byronwatkins2565 1 month ago Raising a real number (>1) to a power yields exponential growth. Raising i to a power yields oscillations: 1, i, -1, -i, 1,... Euler simply developed the continuous (rather than discrete) version of this. Reply @PrivateUsername 1 month ago Wonderful! Reply @weakw1ll 1 month ago 10 seconds in and i understand imaginary numbers better than in all of school Reply @donaldaxel 1 month ago Leibniz, Gottfried Wilhelm Leibniz, 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who invented calculus in addition to many other branches of mathematics, such as binary arithmetic, and statistics. Leibniz has been called the "last universal genius" due to his knowledge and skills in different fields and because such people became much less common after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. His calculus is the one used today, being easily read, whereas Isaac Newton's way to notate is forgotten. Correct me if I'm wrong. Reply @mr_rede_de_stone916 1 month ago Clear, crisp, very cool looking Reply Welch Labs · 1 reply @AkukAkuku 1 month ago Brilliant and illuminative video. Reply @adityamore9435 1 month ago I WISH I HAD A TEACHER LIKE YOU! ❤ from India 🇮🇳 Reply @adikemi5862 1 month ago Best math lesson ever Reply @tomdasilva2060 1 month ago (edited) I expect to buy your book, not because I expect to understand any of it... But just because I am fascinated, by magic... Reply @yabbaso 1 month ago Beautiful Reply @weakw1ll 1 month ago I love the notecards Reply @oliya_b 1 month ago How elegant 😮 Reply @eric3650 6 days ago Yeah dude I totally understood all of this hold me back bro hold me back or I'd show you 1 Reply @IamDarknessS-y4y 1 month ago супер связка, доходность опупительная Reply @SlinkShady 1 month ago (edited) Best explanation I've seen for x^0 =1 was, x^1 / x^1 = 1 because any number divided by itself is 1. As we know this is x^(1-1) = x^0 = 1. Reply @davidwilkie9551 1 month ago Very nice presentation. The adoption of observational Singularity-point orientation-observation equivalence, i-reflection is the same instantaneous superimposed complex plane as projection-drawing perspectives of the 3rdD ..line-of-sight parallels=> dimensionality in a picture-plane @ center of time-timing. (Which everyone "knows" until they are asked how/why) Reply @vivekpanchal3338 1 month ago Your complex numbers series was amazing, to every beginner I suggest your series to watch, amazing, This video is also amazing, With history, logic and math 👏🏻 I hope this series also get extended upto complex analysis, integrals and meanings of residues, I am eager to waiting how you will make videos on such advance topic, ️ Reply Welch Labs · 1 reply @Roman_CK 1 month ago I really liked your video. Well done. Reply @davidnelson2204 1 month ago I have always been confused about why i shinanigans got confined to the unit circle…. Thanks for clearing that up Reply @apollonitro4802 1 month ago Ima buy that book Reply @Serghey_83 1 month ago (edited) Формула эйлера выводиться через разложение в ряд экспоненциальной функции exp(ix) = cos(x) + i · sin(x) Таким образом экспонента мнимых чисел – это периодическая функция. Reply @ianmichael5768 1 month ago Yeah, this is beautiful. True Teacher Reply @RandyKing314 1 month ago i’m totally kicking back and watching this on tv… Reply @isolatsi 1 month ago Some inventions are beautiful Reply @n.a.7723 1 month ago After completing my bachelors in biology and taking pre-cal in high school… I kind of miss having to go through math class and do homeworks on these concepts. Reply @cato451 1 month ago Euler’s number 2.71828 is my favorite of all numbers. Reply @bentationfunkiloglio 1 month ago Spectacular. Reply @gnaneshwaar2594 2 weeks ago Great Video! Reply @ajptrois6671 1 month ago I have e^iπ + 1 = 0 tattooed on my arm as well (among other equations), great video! Reply @ronalddonner3396 1 month ago If I could understand, I mean really comprehend Euler's formula, I could die a happy man. Reply @amirs6472 1 month ago wonderful Reply @softwaresignals 1 month ago Good, now I don't feel so bad for puzzling over how in the holy-hell Euler's Formula could work so well. Most engineers like myself just use it without a proof, and it does work, which is 'proof' enough for us. It is the right tool for the job of inventing all kinds of modern tech, that's for sure. In the evolution of man, in Europe in this case, Euler's works build much of the foundation. Brainiacs. Reply @gabrielbarrantes6946 1 month ago At the beginning of the video you mentioned that everything was part of a more general theory... So all this video can be summarized as cos(z)=(e^iz+e^-iz)/2 and sin(z)=(e^iz-e^iz)/2 and everything follows immediately Reply @scottrackley4457 1 month ago Well. Someone just got some homework. Much better explanation than I got in 94. Of course I had to pause and use pencil and paper. Very nice. Reply @mohammedis-haque8447 1 month ago beautiful Reply @AngeloXification 1 month ago I wish I could be in the room when these mathematicians were reading each others letters. Reply @marcomill4824 1 month ago 1:11 there is a little mistake here, since Leibniz is THE founder of calculus. He discovered first the derivatives 2 Reply 1 reply @williambarnes5023 3 weeks ago (edited) The real problem (pun intended) with imaginary numbers is the name. Had they been called "geometric numbers", people would have immediately realized that they were rising into another geometrical dimension, used them as vectors from the start, and saved everyone a lot of trouble. 1 Reply 2 replies @sfaxo 1 month ago This is amazing. Reply @nigelrhodes4330 1 month ago I wish I had this book like this back in high school when I was having trouble understanding what was going on, I also did not really understand logarithms either I guess, I was searching for the connections and this looks like it would have helped me helped it jell. Reply @z-beeblebrox 1 month ago 1:10 "Gottfried Wilhelm Leibniz, co-founder of Calculus" lmao that "co-" is keeping a tight lid on the drama hiding behind it XD Reply @alexandermikhailov2481 1 month ago If you zoom in good enough, every curve looks like a straight line. That's the foundation of calculus. Reply @bhanuchhabra7634 1 month ago Dear sir, Consider me subscribed for life And thus captivated by your presentation and my mind blown 🤯 Reply @Angor6495 1 month ago mindblowing as usual! Could you make your book also available in the EU? I went on your link to pre-order it but unfortunately that's not possible outside the US. I'd greatly appreciate it! Love your channel, keep up the great work. You're doing mankind a service! Reply @p.s.8949 1 month ago Love how that tatoo even uses the LaTeX default font Reply @vehael 1 month ago If maths was taught like this i believe half of complex problem in mathematics would have been solved by now Reply @wulfbearmusic 1 month ago Really enjoyed this deep dive into eulers formula and its origins! Thank you so much! Do you plan on enabling shipping of your book and poster to the EU someday? Reply @Andreas-yt9wv 1 month ago (edited) Great 👍. Imagine a global challenge to produce the best explanation on such topics. This would be something very positive for all of us! It would Boost our understanding and create a lot of views for the best educater of course 👌😉👍 Reply @heyyyitsjosh 1 month ago What an awesome video. I got halfway through an engineering degree and realized the other day I don’t really understand what imaginary numbers are lol Reply @CookieMage27 1 month ago First time I seen someone sponsor themselves 💀💀💀 -CookieMage#13/27(the red one) 1 Reply @MyEyeofHorus33 1 month ago (edited) 3:00 nope... this just made me remember why I despised maths at school. It's gibberish to me. Reply @uknightedband 1 month ago Very well done Reply @inigoloyola1869 1 month ago 15:01 the line with (2.71828...)^x has x as the exponent instead of z. I get what you're saying, I just thought I'd put this here in case there is a way you can edit your video after its posted. Great video Reply @davidluna5504 2 weeks ago Watching this while drunk is the best experience a Physics student can have Reply @corrdude 1 month ago resonates the part of my brain wrangling particles and waves. Reply @vjdav6872 1 month ago Very cool! Reply @ivella4114 1 month ago i wish math in highschool was taught in tandem with its history and utilizations Reply @ratvomit874 1 month ago (edited) If anything really drives home the close relation between exponentials and sinusoids, it's physics. Any oscillating system will eventually stop oscillating and simply experience exponential decay if sufficiently damped. The only way such wildly different behaviour can arise out of the same system just by tweaking a single parameter is if exponents and sinusoids were somehow related. A similar argument can be made for capacitors and inductors, which on their own exhibit predominantly exponential decay behaviour in a circuit, yet somehow create a sinusoidal oscillator when combined in the same circuit. Reply @andersjjensen 1 month ago I don't know if your channel is underrated or if we are just not enough nerds on this planet! :P Reply @goldowolabi7377 1 month ago This is a good video to show anyone that don't see the beauty of mathematics Really great video Reply Welch Labs · 1 reply @gucajjaguc9438 1 month ago I thought I’ll never found a better explanation for this that the One I san in @3blue1brown … THANK YOU! Reply @sepiar7682 1 month ago this was such a good video! idk if u did say this in the video but it made me want to investigate whether (other bases)^(1/their own alpha) = e which they did which made sense and also was cool! 1 Reply 2 replies @teraxiel 1 month ago Although beautiful, my personal opinion is that Euler's identity [e^pi(i)]-1=0 is far and away the most beautiful equation in all of mathematics. Reply @Vennotius 1 month ago 4 minutes in, and I have to say that I love your style. Reply @ripe_aces 1 month ago I would love to see the surfaces drawn with the colour representing the imaginery component of the function. To the best of my understanding the shape and the colour scale of the surface are both plots of Re(F(x)). (Not sure how visually appealing or legable the surface would be but it would convey an additional dimension of information.) a = Re(z) = horizontal axis b = Im(z) = out the page axis x = Re(F(z)) = vertical axis y = Im(F(z)) = colour scale Reply @tune490 1 month ago I pre-ordered two copies of your book. One for myself and one for my dad :D Reply Welch Labs · 1 reply @krishnendusamajder9804 1 month ago Hey great video. I had a request could you also cover the ancient Indian mathematicians, there is a lot to unpack that no one tries to see. Would be a great video, don't you think. Reply @youssifgamal8545 1 month ago masterpiece Reply @leftbrain965 1 month ago Great video. I had no idea that e, radians, and imaginary numbers were all connected. Reply @dani.i06 1 month ago theres a mistake at 23:12 and every other place in the video that shower that top right note card. to get ride of the base of the exponent (e), you have to take the natural log or ln. In the video, log base 10 was used which would not get ride of the exponent base 10 2 Reply 1 reply @swampwiz 1 month ago (edited) My understanding is that "the most beautiful equation in Mathematics" is what the author says, but with pi as the input value, and with some algebraic rearrangement: e ^ [ i * pi ] + 1 = 0 -> as it has the 5 most important values (0, 1, e, pi, i), the 3 main operations (addition, multiplication, exponentiation), and the equality condition itself. I think a better description would have been "the most beautiful formula/function in Mathematics". I myself have a degree in Mechanical Engineering, and my specialty is vibration analysis, and so the key mathematical kernel in my field is the solution of the differential equation that expresses the balance between inertial & elastic forces. This balance contains Newton's 2nd Law of Motion (i.e. the net force applied on a massy thing results in that thing being accelerated in the direction of the force) and Hooke's Law, which states that an elastic system applies a force to its surrounding that is opposite of the deflected distance. The net is that the differential equation for this has an associated characteristic equation with an imaginary solution, and the final solution is merely the exponential of the solution to that characteristic equation - which is why an elastic disturbance results in mechanical vibration. Reply @NathanHarrison7 1 month ago Beautiful video. Subscribed. Reply 1 reply @therecogniser2122 1 month ago Hello, I'm very stunned when looking at the 3d graph of x^2+1, but I would like to tell you that the graph is more like Re{y}=x^2+1 instead of y=x^2+1 When you expand the b parameter, the x become complex plane while y is still a real axis, not yet a plane. y=(a+bi)^2+1. We can graph this as b=0, then y=a^2+1, a parabola; and as a=0, then y=-b^2+1, another parabola. You can see the paraboloid cut the (a , b) plane at another 2 hyperboles, which means all the point in this 2 hyperboles is the root of equation y=x^2+1, which is wrong. The root of equation y=x^2+1 is +i or -i, cannot be continuous as a 2 hyperboles shape. So I think in order to graph y=x^2+1, you need another dimension for the imagine part of y too, which can only be illustrate in 4th dimension. Correct me if I'm wrong. Thank you. Reply @vascomanteigas9433 1 month ago Next step: The Foundations of Complex Analysis. Reply @LV-ii7bi 4 days ago This is GOLD 🥇 Reply @toddsayles2674 1 month ago Great video. Reply @SylooxD 1 month ago Really great video, from it I finally understood Euler's Formula! But it left me with one open question: You showed that 2*log(-1) is not equal to but only a subset of log(1). Does that imply that the logarithm power rule is not valid for the complex logarithm? (Or at least that it must be used with great care?) 1 Reply @JoseLuisHidalgo 1 month ago Wonderful video, and looks like an awesome book. Any idea if and when it will be available in Europe? 1 Reply @norbertdapunt1444 1 month ago Awesome. Reply @leoarzeno 1 month ago Bravoo! Reply @Asiago9 1 month ago I remember deriving this equation through Taylor series for e^x as an exercise in Calculus 2 Reply @immort4730 2 weeks ago Bernoulli: Wait, it’s all just a Lie Group? Euler: Always has been. 🔫 Reply @timeflex 1 month ago Imagine a spring with one end fixed to the wall. We all know that if we pull it out of its resting position, it will vibrate in accordance with some sine formula. If now we imagine that the force of reaction of that spring flips its sign, then instead of sine the motion of that spring will be described by exponent. And all we did was just a sign flip. Reply @leonhardhoffmann4542 2 weeks ago Absolutely love this Video! Great addition to the original series on imaginary numbers, which i watched multiple times and which also sparked my interest for higher mathematics, as i read it did for many others too! Would love to get your book or poster, but i saw you only deliver within US currently, and i live in germany :/ . Is there any way to get that or other things from you, or should i just join Patreon to support you? Reply @5GWGuerillaFighter 1 month ago Love it Reply @anywallsocket 1 month ago Euler realized n*-1 flips you over zero on the number line, not unlike 180 degree rotation, and likewise n*(-1)*(-1) does this twice and so you end up back where you were, like 360 degrees. Therefore he wondered that a 90 degree rotation would correspond to doing this 1/2 times, ie n*root(-1), which lands you on the imaginary axis at n*i. Reply @neilphilip2320 1 month ago Who needs God when a pencil and paper and thought can produce this level of beauty? Reply @everydayisschool 1 month ago if math was taught like this in my school i would have liked math Reply @hugopristauz3620 2 days ago great video Reply @walternullifidian 1 month ago I can't do math, due to my terrible educational experience, but I can enjoy watching math done by others who can do it well! 🤓 Reply @krishpop-n 1 month ago Lovely video, thanks for rekindling my love for math! I browsed your shop and noticed you had some activation atlases for InceptionV1 from the distill activation atlas. Any thoughts of updating it with a visualization of GPT using circuits? Let me know if you'd like to collab on this! Reply @JoshKings-tr2vc 1 month ago So the math checked out before our own conceptions could be described. Interesting Reply @vimaljain7950 3 weeks ago Nice, bro👍 Reply @ksdnsdkumar1375 7 days ago Expected animation, we get man writing maths forumulaes on piece of paper Reply @askcaralice 1 month ago the origin of e definition is interesting, i did not know that, but using degree measures in complex algebra is something absolutely cursed and unforgivable Reply @KalkuehlUncut 1 month ago Paper published in 1747 and Bernoulli dying in 1748 is a corrilation we shouldnt ignore. Reply @mikejones-vd3fg 1 month ago (edited) Some might find beauty in complexity, some find it in simplicity, which means there isnt an absolute beauty, whats that cliche saying.. ahh yeah beauty is in the eye of the beholder. Reply @theoneeditor399 2 weeks ago pi : Get real i : Be rational e and (-1) : Join us and we will become nothing PI AND I RUN SCREAMING IN TERROR Reply @isuckatthisgame 1 month ago As an engineer I can confirm that without Euler's formula there is no system analysis. As a matter of fact, Euler also discovered an elegant way to describe discrete signals from continuous domain. Euler is one of the most important people who ever lived. 2 Reply @Simons_Valere 1 month ago Great stuff 👍🏻 Reply @jojojorisjhjosef 1 month ago Beautiful ending. Reply @imacmill 1 month ago I have to say that I kinda got the butterflies in my stomach watching this, as I was somewhat able to grasp the analysis given. But there's no way in heck I could regurgitate any of it on demand, except for the final equation, of course. Now...what has mankind done with this equation? Reply @crp5591 1 month ago Ooooof... It's videos like these that drive home just how immeasurably dumb I am. Great video.. I just wish I understood any of it. Reply @mrhassell 1 month ago Couple of real "GEMS" there... Ross Honsberger's two books, AMS - American Mathmatical Society, Mathematical gems by Honsberger, Ross, 1929 - would have been the go to, before your book went places, he was simply technically never able to! Reply @yoavboaz1078 1 month ago The long awaited sequal Reply @forthrightgambitia1032 1 month ago Perhaps it would have been fair to give De Moivre a shout out here 10:45. 1 Reply @stephencole9289 1 month ago I knew Eulers equation but didnt realize this is how or why he came up with it. Reply @karkaroff1617 1 month ago we're so back Reply @electrofly23 1 month ago awesome graphics! great presentation. now if you'll excuse me I have to go hit that subscribe button 🙂 Reply @thenightjackal 8 days ago His Euler's Identity tattoo is upside when from his perspective. I would just cut off my arm if that were me. Reply @matthewkendrick8280 1 month ago One time I tried to ask Siri something about Euler, only for it to not understand and it thought I said oiler. I had to pronounce his name like yew-ler for the stupid robot to understand. 1 Reply 1 reply @therealpils 1 month ago That was the most beautiful explanation of gobbledygook I've ever witnessed, thank you. ps. am, clearly, NOT a mathematician. pps. does you book have lots of pictures? Reply @nopenope9945 1 month ago I was with you there for a while, just listening, then could no longer follow that way alone, I needed to see it. And that's when it happened. I couldn't find my phone. I was in a panic. My chest got tight and I was like - sq foot of i... oh NOOOO. But then I found it and everything was ok. Don't do that to me again, bro. Reply @TheBillzilla 1 month ago I got lost a bit more than a minute into it, right to the end, but it's no doubt all extremely clever. Reply @user-cg7zn8ey5k 1 month ago Phantastic video with great insights! Thank you for sharing. Btw. are the formulas written by using LaTeX? Why do you use \lim but not \sin, \cos, \tan, \ln, \log, ...? Reply @bluebonnet 1 month ago "I'd like log(-1) eggs, please." Reply @mtheory85 1 month ago Euler: "Ackshually..." Reply @JuBerryLive 1 month ago Thanks for that video. I'm still trying to understand if imaginary numbers are... "real". In the physical sense. I know they are a construct, but they have so much predictive power. It makes me want to believe that our physical world is purely mathematical. It blows my mind. Reply @michaeljames5936 1 month ago I'm watching this a little hungover on a Saturday morning. Anyone remember 'The Open University' lectures on BBC2? Many's a morning, you'd watch along, amazed that you were able to understand it, then you slightly ignore one or two details, cos you're not quite getting them, and before you know it, you're just listening to a man talking. Ditto- this vid. (I've so missed maths since I stopped studying it. I had withdrawal effects when I quit, and I still get the odd craving, but I doubt my brain could even get me back to the level I stopped at, never mind take me further.) Reply @jamesdonaghy9143 1 month ago I found that pleasantly soporific. I have only the faintest grasp of some tendrils of your thought arc and find myself nestled like a weakling on a branch of your muscular mathematical tree. Reply @sikkepitje 1 month ago Very nice table BTW Reply Welch Labs · 1 reply @seifyk 1 month ago This is so good i was expecting to see little blue PI characters from time to time. Reply 1 reply @chaos.corner 1 month ago Can't believe you didn't mention Euler's identity. Reply @aisolutionsindia7138 7 days ago imo its not really a formula but a shortform to capture a function which has the following property f(x+y)=f(x)f(y), nothing much changes in math without it Reply @phishbutter 1 month ago You know how Garfield the cat became famous for hating Mondays? That’s how I feel about this video. Reply @ecdavek230 1 month ago wow ... that was nice Reply @mandeesman7889 1 month ago I haven't made it that far in math but thank you so much I did not know that logarithms are another way to express exponents😢 Reply @Valeriy7D0 1 month ago 5:12 This integral isn't the area of the grayed sector, it's the area of a "slice" of a 1/4 of a circle , I think Reply @roomcayz 13 days ago That's insane that not so long ago I had understood and knew all of it, but forgot it all once I passed the exam, lol Reply @KoushaTalebian 13 days ago Why are we still talking about complex numbers as "imaginary"? It is simply a rotation operator of 90deg. The reason why i^2 = -1 is purely because of the Cartesian coordinate system we use. Reply @shoobidyboop8634 1 month ago I like this. Reply @MerlinZuni 1 month ago This is so over my head. Beautifully presented and explained. I am just missing some foundational understanding. Reply @EkShunya 1 month ago mind blown Reply @byrongibby 1 month ago Envious of those learning mathematics (for the first time) today, as good as my textbooks were/are they don't come close to this :) Reply @manfredbogner9799 1 month ago Sehr gut Reply @frzferdinand72 1 month ago I've thought about getting that exact same tattoo! Reply @ByteMe1980 1 month ago Love your videos man but your vocal fry is realllllllly strong man. borderline bearable Reply @kharnakcrux2650 1 month ago I remember seeing these manifolds, that unify trigonometry and hyperbolic functions. They're all the same. And the symmetries involved, Srinivasa Ramanujan..... And his q series and mock theta functions.... String theory critical dimensions.... See where I'm going with this? Reply @skylerthacreator 1 month ago so sick Reply @letmewatchmyshows 5 days ago This is good. Reply @firecrafter28 10 days ago 4:39 “💀💀💀” - Euler Reply @drakouzdrowiciel9237 1 month ago thx Reply @lupevelez5723 3 weeks ago Gottfried Wilhelm von Leibniz got me though school. To this day i love the readings of The Theodicy or The Monadology. Reply @feelfree.1 1 month ago I wonder what Euler would say about Quaternions 1 Reply 1 reply @primaryesthethicinstincts4832 1 month ago This is more of a numerical approach to the formula. Reply @cyboticIndustries 1 month ago i just knew I'd seen that thumbnail long ago someplace ...... 😀 Reply @Magnasium038 1 month ago Huh, so logarithm is technically not a function when we include negative domain and complex range. Today I learned. Reply @darkfool2000 11 days ago It's not Brigg's equation. That equation is the definition of e, and Euler and Bernoulli discovered it first. That "zooming" in method is also the fundamental basis of Calculus discovered by Newton and Leibniz. Brigg was just the guy who decided to make this massive table of base 10 logarithms that could be used by anybody. It's less discovering of new mathematics and more making mathematical tools that are readily accessible to more people. I know it's easier for people to understand than the actual way Euler derived his formula, which was from Taylor series expansions, but it's still misleading to present it without the appropriate asterisks. Reply @mjifi 1 month ago Hope your book does well 🎉 Reply @THEADVISOR-OneFromBelowAbyss 1 month ago I just don't belong here. I know nothing that is going on, but i'm terribly enjoying it 2 Reply @poweruser6995 1 month ago Yuler is how I pronounce Euler. Heard Oiler first time! Reply 1 reply @edmondthompson7525 1 month ago Terrence Howard is punching the air right now! Reply @GokulVijai-y6l 1 month ago Can you create a similar explanation video for Fourier transformation and Fast Fourier transformation. 1 Reply 1 reply @ivocanevo 1 month ago (edited) A question for physicists or mathematicians reading this. The final graph showing the relationship between log/exponent on one plane and sin/cos on the other seems profound. It feels like it could hint at new ways of solving problems, adjacent to exponential infinities, limits of oscillations, maybe emergence of dimensions even. I'm probably making too much of it. I guess my question is: are there some real examples of how this relationship sheds light on physical reality? (In the graph I see a short and a long wavelength on two ends, connected via an infinite exponential curve and rotated through an additional dimension. I'm reminded of things I don't understand: curvature at event horizons, ER=EPR holography or Penrose's intuition of Conformal Cyclic Cosmology. Geez, I hope someone takes me seriously enough to answer, even to put me in my place. I don't think ChatGPT is going to know this one.) Reply @pranavsagar1338 1 month ago What was the opening song! It was truly astounding and it helped drew me into the rest of the video 😊 Reply @VietVuHunzter 1 month ago Hear "math": of course it's Euler again. Reply @ReidarWasenius 1 month ago Thanks for another GREAT video. 😊 At 9:16, the sum angle looks to be very close to 90°, so: 55° doesn't seem right at all. Hmmm....checking, reveals that the addition of angles should be ca 18.43° + 63.43° = 81,86°, right? Reply @ldmtwo 1 month ago I never cared so much about a math video. I'm an engineer. Reply @Alex-js8pu 1 month ago Amazing - do I just like big words? I'll watch again.😁 Reply @easterndundrey 1 month ago you're a genius Reply @MissionSilo 1 month ago Books looka great Reply @williamarcor251 1 month ago That bit at the end is funny, I've always thought if I ever got a tattoo, it would be Euler's formula. 2 Reply @arthurfrost9004 1 month ago My love for math had died on the day my fourth grade math teacher insulted me. Your video sparked some curiosity but I still hate math. Reply @Kgsi424 1 month ago The inconsistencies found in the beginning sounds a lot like the story of Plato’s Cave 😂😂 Reply 1 reply @santiagomartinez3417 1 month ago Tristan Needham wrote a book with excellent reviews. Reply Welch Labs · 1 reply @jackdan1811 1 month ago Videos like these are why i pay for my internet Reply @gabrielmonopoli7994 1 month ago Amaizing graphics software you are using to show it!!! Please could you tell the name of it? Reply @mauriceledoux3009 1 month ago Great video! Its nice to see the historical background surrounding euler's formula. After watching Grant Sander's summer of math during lockdown, he mentioned the difficulty of finding an intuitive meaning for imaginary exponents (specifically for i^i). I find it interesting that you mentioned here that Euler had similar difficulties. I think I found a very intuitive explanation for what these concepts "mean", would be happy to discuss since I think it would make for a great video. DM me if youre interested! Reply 1 reply @peta1001 1 month ago (edited) Yes...I passed all the math exams while obtaining an electronic engineering diploma...only to conclude that I was able to repeat (apply) the math rules by hart. However, one simple problem, that always put my colleagues in fight, was the definition of multiplying two negative numbers. If multiplying is short addition (repeated addition), how is it possible that -5 multiplied by -2 equals +10 (which, on the number line, is 15 units/points away from the multiplicand -5)? Please, comment this simple question... make me trust that main stream math is not built on convenient truths and methods, which may be essential to not understanding much about the universe, quantum mechanics etc. Reply @Eumanel12 11 days ago This is beyond me. Maybe one day I'll understand ot Reply @TheDivergentDrummer 1 month ago Dude, I hate math and this was amazeballs. Seriously cool maths right there. Reply @ルクミ-w8m 1 month ago 14:53 That moment when you are pro science but you start to think there must be some form of higher beings who invented "e" and put it everywhere Reply @lycancrystal 1 month ago Why did you take the conjugate of 2^bi and 2^-bi , so ur using the Euler’s formula in its own derivation 💀🗿🗿🔥🔥🔥, we be cookin with this one 🔥🔥🔥🗿🗿🗿 Reply @TheArtOfBeingANerd 1 month ago I need mathematician trading cards now Reply @IanZainea1990 1 month ago This video oddly made me realize that negative numbers are also not real. You can't have a negative amount of a real thing. Even electrons being "negative" and protons being "positive" are just what we call them. They aren't actually negative. Just opposite charges. I can't have a negative number of fingers for example. Or a negative number of atoms. Or a negative number of eyeballs. So on ... 1 Reply @nufosmatic 1 month ago I am going to have to watch this about three more times before my crufty old brain is going to wrap all of the way around it. Reply @LupisLight 1 month ago (edited) Question, Euler's formula tells us how to evaluate a real number raised to an imaginary exponent, but how do you you handle an imaginary base raised to an imaginary power? For instance, 4^2i is on the unit circle (magnitude of 1), but 4i^2i is NOT. in fact, punching in an imaginary base to an imaginary power on my calculator seems to always result in a real number, but I have no idea why or how to evaluate such a thing on paper. Reply @edbail4399 1 month ago So it takes more than 200 mathematicians 50 years to understand the square root of -1 body-blue-raised-arms Reply @talatdhk 1 month ago For all those who didn't understand and grasp what this is all about, e^(2πi)=e^0...😢 And for those who got it, e^0 = +1😊 Reply @orterves 1 month ago Be sure to post a reminder when your book can be shipped outside the US, the current page supports a US address only as far as I can tell? Reply @oluwatomiwaamosu2043 3 weeks ago Ok now I gotta love maths Reply @tomholroyd7519 1 month ago (edited) Imaginary truth values are valid. #RM3 Complex truth values arise as an extension field of Z2 the same way i = sqrt(-1) creates the complex numbers from x^2 + 1 = 0 --- The Liar Paradox, in Boolean Algebra, is x*(x+1)=1 (mod 2) "x and not x is true". Z2/(x^2 + x + 1) is F4, with two new "imaginary" truth values, Both and Neither, which solve the Liar Paradox. Reply @kapilchhabria1727 1 month ago Could you do a video on fractional derivatives? Reply @bradhayes8294 1 month ago Euler's like "Hold my beer". Reply @ashketchum4953 1 month ago (edited) 18.4 + 26.6 = 45.0 (not 55.0 as suggested at 9:17) Also at 9:37, multiplying a vector by its complex conjugate returns the magnitude of the real component, not the magnitude of the entire vector. Your explanation alone does not satisfy that 2bi lies on the unit circle. 1 Reply @drslyone 1 month ago "This video is sponsored by me" And I'm going to visit your sponsor. Reply @realcygnus 1 month ago Nifty AF! Reply @Grateful92 1 month ago Wow^ix= subscribed❤ Reply @JCCook205 1 month ago (edited) 23:21 πi, Captain! OOOOOOH who live in a pineap.... sorry got lost there for a second. Reply @rsn8887 1 month ago (edited) Excellent video, thank you! I have a hard time understanding your 3D surface plots, mostly because you didn't put any axis labels. You have a function that relates a complex number x to another complex number f(x). How do you display that as a 3D surface? In your plot, where is the imaginary part of x shown? The paper surface only shows Re{x} and Re{f(x)}, I think, and Im{x} is plotted on the axis coming out of the page. But where is Im{f(x)} plotted and what is the meaning of the colorful surface? Reply @CubisticWhale 1 month ago "Only shipping to US addresses currently" was a tough line to read on the book order page. Is there a chance the book will be shipped internationally? 1 Reply @YMandarin 1 month ago Analysis course flashback Reply @deniskhafizov6827 1 month ago Using approximate values intermingled with exact ones without specifying which is which - this is not math anymore, you've turned it into pampering with a calculator. Reply @PolyEthylenTerephtalat 2 weeks ago (edited) 5:24 „quite complex“ lol Reply @davidherene6365 1 month ago All I understood from this video was the word "Euler" which sounds like Oiler. Like Oilers Reply @jesterlampoon-t1j 1 month ago ✋Math🤚 1 Reply @lolilollolilol7773 2 weeks ago (edited) The most beautiful equation is Euler"s identity: exp(i Pi) + 1 = 0 Reply @atahirince 1 month ago one day want to watch math without being lost Reply @dowesschule 1 month ago I don't get the graph at 25:37 . cos b is supposed to be the real part of the result if a is 0 — but it is at a -> infinity. Which it has to be, because at a -> 0 the b-axis has to be 0 too, otherwise there would have to be a change in e^a as well. Because the manifold's intersection with the plane gives the graph, not it's projection, right? 1 Reply @mattshannon5111 1 month ago Great video! Your latex looks unnecessarily ugly though because you use log(x) instead of \log(x) and cos(\theta) instead of \cos(\theta). Latex interprets this as l o g(x), hence the funky looking expressions, particularly noticeable in expressions like "i s i n(\theta)" 1 Reply @beofonemind 1 month ago ok cool. I understood 0.1 % of this!. Thanks to all you geniuses who study this so then we could build things to lessen suffering. Reply @oraculox 1 month ago Is explanations like this that make me aware of why I always flunked math, the virtuality of numbers makes my system stall hhaha, I had to use the .75 speed playback 🙇‍♂🤦‍♂. Reply @Zantsui 6 days ago When the book releases, can you work on international shipping? Id pay for it to come to Australia. Reply @zyzhang1130 3 weeks ago Why I feel like I’ve watched this video years ago Reply @Karel8X 1 month ago After many semesters of math at a technical university, I still hate math. Why? Mathematicians think differently than us normal people - they can understand math, but they can't teach it. Reply @SasaMrvos 1 month ago Will you be shipping your book outside of US? I wanted to preorder it, but EU countries are not on the list. 1 Reply @johanneskingma 1 month ago youtube's algorithm is good on you. How did you do that? was it expensive? is revenue going to Alfabet? or is this all adware generated. I'd sponsor you but having this in my feed makes me wonder how much Google makes hare. Reply @algebraicoo 4 weeks ago Hey I wanna do a video in spanish about this very toppic, could I take the 3D animations from your video? I would reference your video 1 Reply @d1tnhauxa0rau 1 month ago it feels like I saw the intro animation somewhere ele Reply @peters972 1 month ago I stumbled on FFT (fast Fourier transforms) when trying to produce spectra of stock price streams in a financial application. It uses imaginary numbers to plot wavelength and magnitude as you will know. Of course Euler had quite a fascination of compounding in finance. I was wondering if there is some further interesting connection between binomials, e, Fourier transforms and the beautiful equation you described above. Also, just wondering if there is also a link to Schrödinger/Einstein since there is a mapping from waveform to “partical” if I may use some poetic license to call a compounding number a particulate, lol. Reply @kenjiesensei 1 month ago Damn I really hope you ship international Reply @abhyudayshardulsingh5159 3 weeks ago Hey there watched your imaginary number series 2 years back and I am back for this video very excited for your book do you deliver to india or can I buy it through Amazon? Reply @노진호-h4f 1 month ago 수학에서 가장 아름다운 식이 저렇게 나온 거구나! Reply @kingplunger6033 4 weeks ago Haven't I seen this before somewhere Reply @mathieud5594 1 month ago erratum: 18.4 + 26.6° does not equal 55°, but 45° Reply @FutureAIDev2015 1 month ago 1:12 I'm curious why they wrote in Latin since the Roman empire had fallen hundreds of years prior if I remember right 1 Reply 1 reply @radmehrhakhamanesh6816 1 month ago I hope That guy who said “I made up the root of x^2+1=0” is burning in hell Reply @TheEndermanMob 3 weeks ago Here in Brazil if you write log without specifying the base we assume base 10. So for much of the video I was confuse until I remember the euler has it own number so you must be talking about log of base 'e' Reply 1 reply @leonhardhoffmann4542 2 weeks ago Addition: Do you plan to enable direct video money donations on Youtube? Have seen on other channels/videos, that there were comments which also gave donations, i believe many people would be willing to support you that way (possibly me included) Reply @evasuser 1 month ago 22:18 table of properties. Reply @thedayb4tomorrow 1 month ago Also the ith root of i is real 🙂 (or more accurately, all infinitely many ith roots of i are real and take the form e^((2n+1)*pi/2) where n is any integer) 1 Reply @betims 3 days ago Could you do a video on operator algebras please? Reply @paulbabypfelski6193 1 month ago Can you substitute sohcahtoa for the Power Identity? Think equilaterals with motion/Orbit variables! Reply @Engr.kennedyAnthony 2 days ago (edited) This guy is funny am wondering how your wife must be very calculative because you seem to be a tough guy mentally! Thank you. I just followed you. Reply @filippocontiberas 1 month ago 23:24 - Is there a rule that tell us which ( between logarithm or squaring) has higher priority? In that case: taking out the 2 of squaring first or squaring first and the logarithm after? I think must be an unique correct execution in any math expressions. Reply @rshtg2019 1 day ago i cannot believe i was right Reply @MrAndrew535 1 month ago Not "math" but "Maths", the latter and not the former, being an abbreviation of the word, "mathematics". One wouldn't say, "mathematic" would one? I wrote a forty-thousand-word paper in which I asserted, categorically that, "in all things, Language Matters". Reply @FosterNelly-f5p 4 days ago 656 Vandervort Mountain Reply @moseskim3942 1 month ago Are you a PhD? I can’t understand the math but appreciate the concepts and storytelling! Reply @Valeriy7D0 1 month ago 3:35 if the derivatives are the same, shouldn't the left side of the graph go down-left instead? 1 Reply @spireneusz 1 month ago Angle for 1+2i is around 63,5 deg not 26.6 as you put in video @9:15 thus 7.07 have angle of almost 82 deg Reply @Fetrose 10 days ago If you don't mind, could you please let me know which software was used to create the amazing 3D animation? Reply @lisaprince1313 1 month ago damn this is good Reply @tony0000 1 month ago I have a very limited background in math. Would it be fair to say that Euler's formula, which defined the raising of a number to an imaginary power, was accepted because it allowed Brigg's formula to be extended to imaginary numbers, much like the equating raising a number to 0 with 1 was accepted to preserve the additive property of exponents (if that's what you call it)? Reply @baxtermullins1842 1 month ago The same is true of the Sigi-Bode and root locus. Reply @jayosborne 1 month ago I am so happy and so proud that there are people in this world that smart. But my brain hurts and I’m gonna go find a cat video. Reply @Soupie62 1 month ago The models in this video are great - but screens are only 2D. Is it possible to make a 3D model? A combination of wire and epoxy resin might make a free standing version. Failing that, lasers making 3D objects inside crystal were popular, a few years back. Reply @davidwilkie9551 1 month ago From a training in holography-quantization-> Numerical whole message unity-connection instantaneously, by default, which every body knows in the Eternity-now superposition of Being Here Now, ie .dt zero-infinity sync-duration Singularity positioning is entangled all-ways all-at-once here-now-forever. Reply @Npvsp 1 month ago Can we stop to call it an equation?! It’s an identity! 1 Reply 1 reply @jasonlaguerre923 2 weeks ago Let me know when you’ve started shipping to Canada Reply @Evoleth 1 month ago 9:49 2^3 × 2^(-3) also equals to 1. But that doesn't mean the magnitude of 2^3 is equal to 1. So how could 2^(bi) × 2^(-bi) = 1 mean 2^(bi) has a magnitude of 1? Reply @tablebook-dg6vh 1 month ago Dan carlins episode of maths Reply @rafbambam 1 month ago (edited) Hi, first of all, thanks for this realy great video. But I still have a question. The exponetial graph grows to infinity, but a sin- or cos-function doesn't go to infinity. So how can a "shadow" of a finite function be infinite? In other words, how can a sinfunction that is crossing the real plane have an infinite crossingline? Greatings from Belgium. Reply @clintflippo917 3 weeks ago I watch videos like this, with no understanding of calculus... agreeing with everything, acting like i understand... but i dont... i truly dont. Reply @tonychinnery 1 month ago Defining i as 'the square root of -1' is wrong. It should be : 'i is one of the two square roots of -1'. This is what got Euler himself into trouble when he wrote: sqrt(-2) x sqrt(-3) = sqrt(6) . The fact is that sqrt(-2) is not a number, its a variable with two possible values. The convention for positive numbers to choose the positive square root cannot apply to imaginary numbers, as they are neither positive nor negative. Reply @michaelgrant6332 1 month ago That was great. You didn't show its most well known result - when θ=π. Reply @saturnslastring 7 hours ago I feel like I watched this video a very long time ago... Reply @diribigal 1 month ago This video is an excellent summary of some important history and mathematics. It also bugs me that all the functions like "sin", "cos", "log", and "ln" are written in italics like they're products of variables, instead of upright like the (La)TeX commands \sin, \log, etc. For the video it's just a minor thing, but for the poster (and book if it displays the same way), it's a slightly bigger issue. It sounds like it's too late to change the poster, but I hope the book displays differently/can be changed. 1 Reply @Quentyn73 4 weeks ago "This video is sponsored by ME" = immediately subscribe Reply @rafael_tg 1 month ago Will you release a digital version of the book (2024 version)? Reply @methatis3013 1 month ago You need to be very careful with this. Usually, we need to mention whether we are treating log as a function or just as a way to express which x satisfies the equation e^x=c where c is some constant. This is a problem when working with complex logarithms. We need to be very careful with which rules (that we usually take for granted in real numbers) actually apply Reply @ПараноидныйСиндром 1 month ago n < (n+1) is True for all n = {0,1,2,3....N} a^n = a ---> n = 1 a*n < a*(n+1) a^n < a^(n+1) 1) Let: n = 0, a = 0, Assume: a^0 = 1, a^1 = a 0^0 < 0^(0+1) ---> 0^0 < 0^1 ---> 1 < 0 - Contradiction!!! 1 not strictly less than 0. 2) Let: n = 0, a = 1, Assume: a^0 = 1, a^1 = a 1^0 < 1^(0+1) ---> 1^0 < 1^1 ---> 1 < 1 - Contradiction!!! 1 not strictly less than 1. 3) Let: n = 0, a = 2, Assume: a^0 = 1, a^1 = a 2^0 < 2^(0+1) ---> 2^0 < 2^1 ---> 1 < 2 - True. Reply @slep5039 2 weeks ago But why is the complex conjugate of 2^bi = 2^-bi? The explanations I've seen use Euler's formula, which would make this a bit circular Reply @DemonetisedZone 1 month ago 2⁰ = 2 I don't understand! 2² =2x2 2¹ =2 multiplied by itself once 2⁰ = not multiplied by itself it just stands alone as 2 Is that it? 1 Reply 1 reply @MrKydaman 1 month ago Please don't let Terrence Howard see this video. 🤯 Reply @TroyRubert 1 month ago Euler was too op. Reply @chipsystems 2 weeks ago So, 9:18 has a bunch of the angles wrong, and it was driving me nuts. I don't know if I'd known before about multiplying in polar notation by multiplying the magnitude and adding the angles. That sounded so wild to me that I had to test it. And, my numbers weren't working out. Anyway, it does seem to be true (can multiply in polar notation by multiplying magnitude and adding angles... I'll have to investigate why that is later). But, in the video, the vectors should be 3.16@18.4 (correct) * 2.24@63.4 (incorrect, the complementary angle) = 7.07@81.8 (incorrect). Reply @FredZiegler63 1 month ago @time 9:17, the blue vector is shown as having an angle of 26.6 deg. This is clearly not true - the actual angle > 45 deg. Also, 18.4 deg + 26.6 deg is clearly not 55.0 deg Reply @robertbox7666 1 month ago Katherine used this in Hidden Figures Reply @shannontaylor1849 1 month ago I thought I was gonna see some pretty pictures and stuff. #high school algebra. Reply @komodesu-v6g 1 month ago cool video Reply Welch Labs · 1 reply @imacmill 1 month ago If we have 'i' to represent sqrt(-1), maybe we should have an 'I' for 0^(-1). 😁 Reply @1088lol 1 month ago 5:15 ghosts bro Reply @piradian8367 4 days ago At 5:22 how did you get log expression with complex args there for that integral? I’ve got this expression 1/2 [x*sqrt(a^2 - x^2) + a^2 arcsin(x/a)]. Reply @LittleMissFired 12 days ago Number the quadrants=top right 1. Reply @kapilchhabria1727 1 month ago Would it be accurate to say that since the formulation of calculus is demonstrably correct, Euler’s representation resolved a seeming inconsistency, and only this representation resolves the inconsistency? Reply @ArunJayapal 1 month ago Website down at 15th Aug 2300 ist. Interested in getting the book Reply Welch Labs · 1 reply @paolooppezzi830 1 month ago at 9.19 you say ( and write) that 18.4°+26.6° = 55° . The correct sum should be 45°. Also in the graph all polar angles look wrong: if measured from the x axis how can 26.6° looks bigger than the bisectrix. How come? Reply @laszloszilagyi8788 2 days ago (edited) I like your presentation. (I'm an electrical engineer.) At 09.30 time point of this lecture we can see that 2^bi and 2^(-bi) unknown functions are conjugate of each other. Can anybody help me to understand how I can conclude this from the previous facts in this video ?? This would be the key condition that they { 2^bi and 2^(-bi) } are on the circle of unit radius as I see. e.g: I know that finite polynomials with real coefficients P(bi) and P(-bi) are conjugate pairs because of the relationship between four basic operations and conjugation. But in this case at this moment the function of power of 2 is unknown so I can't see why { 2^bi and 2^(-bi) } are conjugate pairs ? Reply @rocksmashgaming4199 1 month ago (edited) What an amazing video it has been pleasure watching it. great explanation, but it will be great if someone can explain where can we find sin b plot,I think it (sin b) can projected on x-b plane is it?? At 25:38 Reply @jumpsidx 1 month ago damn its all staring to make sense Reply @iznasen 1 month ago only if I watched this video 15 years ago Reply @CristiNeagu 1 month ago (edited) 2:47 It totally does make sense for negative exponents. If a positive exponent means multiplying the number by itself, negative exponents mean dividing the number by itself. 2^3 = 2 * 2 * 2. 2^(-3) = 1/2/2/2 Reply @alexandrudumitru3084 1 month ago Please do not aproximate the magnitude (absolute value) of a complex number and the angle from the x axis ! White the magnitude as a irrational number and the angle using arctan() function ! It's much more easy to understand the polar form in this way. Reply @theseusswore 1 month ago (edited) i love the channel, the content and everything but..$20 for a poster? Reply @georgehelliar 1 month ago I think the answer will be about 4. As in, im going to need to watch this about 4 times before i get my head round it Reply @hans934 1 month ago How did you create the start of the video where you pull out the graph? Is this done with Blender or simular? Reply @ASDasdSDsadASD-nc7lf 1 month ago (edited) For anyone curious, One of the things mathematician Euler applied the math too was calculating the specific energy of liquids and is the origin of the word we now know as "oil", a corruption on Euler's name. In fact the original device to extract oil from the ground was called an Euler's rig and the men operating it were called "Euler's". Over time it got transmorphed into the word "oilier" and the liquid became oil. 1 Reply 1 reply @JoelEngineer 1 month ago 3B1B collab soon? Reply @kapilchhabria1727 1 month ago I’m curious, did Briggs begin with the limit representation of exp(z) and then determine that his 32 orders of scaling was adequate before constructing the log tables? Reply @hamarana 9 days ago How about squaring up the roundness of the planet with 1 square meter ceramic tiles, disregarding mountains? How many 50cm2 tiles would it take? what would the angles between two sets be.. wow a perfect ground.." so if we can imagine this level of tiny space, than if we can add a layer of triagle energy to the surface of every atom in the universe that way we can make the universe grow, inflate, without anybody feeling a thing, making the small get more space ? Then imaginarily we can make the ininfinitesimal grow infinitely smaller? We can grow up and down? Reply @alanbregovic8889 1 month ago mayb i should revisit 2nd grade and onward after 40y Reply @rid023 1 month ago (edited) 9min :33sec 2^(√-1) * 2^(-√-1) = ? * 2^(-√-1) 2^0 = ? * 2^(-√-1) 1 = ? * 2^(-√-1) What next ? We don't know that 2^(-√-1) is conjugate to calculate moduo. Reply @DeniseHaley-z7x 1 month ago 857 Bode Meadows Reply @Unknown-mf4of 1 month ago You have some obvious math errors at 9:15. In yellow, 7.07 at an angle of 55 degrees should be 7.08 (minor rounding error) at an angle of 45 degrees (a rather large angular error). Reply @yongmrchen 1 month ago A footnote for this video is “Mathematics is discovered.” 1 Reply @h84goD 1 month ago i failed math in german university but complex numbers dont let me go Reply @benisrood 1 month ago (edited) Thank you for sharing this with the world. This is so well done... but... 😢 degrees? not radians? Edit: Ahh, you were holding it bsck for the reveal? There's no need, just use radians. 😊 Reply @AnthonyKully-o4v 1 month ago Wolf Ford Reply @danielcohen4839 1 month ago Aliens! Reply @harshitgupta7987 1 month ago 5:25 can anyone tell me whats the real integral of what Bernoulli was solving 2 Reply 1 reply @1guycooperful 1 month ago 00:09:19 Great video, but t’s 45 degrees not 55 degrees, right? Reply @coryloveless6526 1 month ago What do you use to make graphs like those at 3:00? Reply @ahmedfahmy6525 1 month ago Wow Reply @347573 1 month ago how to receive the book in Italy? (the website seems to allow just USA)... isn't viable an "easy" (for the buyer)Amazon distribution? Reply @lukaszligezinski5368 1 month ago please figure out the way to ship that book internationally. I would love to have it so nicely printed, pdf will not do the trick 😞 Reply @luiz5411 1 month ago Please add a paypal option for buying your book. Reply @anakimluke 1 month ago So, aside from the fact that Euler's solutions are useful and play nice with existing maths.. why did those stick through time? Is there something else fundamental about those that I don't understand, or if someone were to come up with a different definition that is useful for another set of maths but clashes with the current ones it would be equally as valid? :) Reply 1 reply @alexanderbergmann4405 1 month ago is there a plan, to ship the book to germany or europe? I really like to have, but don't live in the US :( Reply Welch Labs · 1 reply @dannil9878 1 month ago My brain hurts Reply @debilista 3 weeks ago I had this all in my first semester of engineering. To be honest i forgot everything XD Reply @SSN4781 1 month ago wow Reply @bricology 1 month ago You lost me at "Hello, . . . " Reply @pedrammehbudy 1 month ago Can anybody explain why 2^-ib is conjugate of 2^ib? I mean it definitely is but not with the historical explanation that you were going through. I mean at that time it would need a proof. Just because law of exponents shows that their multiplication is equal to zero does not prove that they are conjugates it just shows they are reciprocal which is obviously true. Reply @spacemanwillie 1 month ago Somehow, this 'zooming in' and 'zooming out' technique to reveal an answer usinf calculus FEELS like it could be connected to the wave function collapse problem in quantum mechanics. Has anyone ever made that connection? Weird intuition that just popped in my mind while watching. In other words, the resolution of measurement apparatus seems to matter in the perception of the existence of reality, and it may be just a logical math problem afteral? I may be completely off onto an unrelated tangent though... 😅 Reply @krwada 1 month ago You tattooed Euler's formula on your forearm??? 2 Reply 1 reply @nopenope9945 1 month ago Oh, I love your different sizes of cards for chunks of explanations. It will be hard to cary around but I need to do do that. It makes an idea have its own area shape and size and then I can just touch it and know it's one of only 15 or 20 x sized squares and can therefore only be one of those proofs or equations. Mathnasium is great but they rip brown paper and write with sharpie and I want to die. Reply @Dusk_Shade 1 month ago An excellent example of the fact that sometimes, math is just plain ol' made up. Reply @lexsouy 1 month ago É uma pena não ter áudio em português. 1 Reply @RolandMeroy-r2q 5 days ago Dickinson Vista Reply @tibors6986 1 month ago e^i 2π=1 Reply @rykerwatt7499 1 month ago At 14:58 2.71828 is to the power of x when it should be z Reply 1 reply @micklogan5489 1 month ago Can someone please tell me what that cloth hat thing Euler is wearing in all his portraits is? Reply @NoraTim-z1w 1 day ago Larson Road Reply @micknamens8659 1 month ago 3:34 If f'(-x) = f'(x) then f(-x) = - f(x) Reply @BenInSeattle 1 month ago Is it a typo at 15:00 where the variable z changes to x and then back in the second to last equation? I hope so because I don't understand it otherwise. Reply @quyentruong5272 1 month ago at 3:42, shouldn't it be symmetric across y=x if the derivative (slope) is equal at x and -x?= Reply @proof6930 1 month ago Its a nice explanation, but it uses several concepts that you did not justify before using them. You didn't explain why imaginary numbers are perpendicular to the the real numbers or why you can calculate the magnitude of a complex number. You also used polar forms of the number and the concepts of adding angles when multiplying, but did not explain why that is valid. Euler's formula can be used to show these properties, but you used to properties to prove Euler's formula. I think is it easier to show how to arrive at Euler's equation by using a general formula to calculate the square root of a complex number. Use that formula to calculate successive square roots of -1. Reply @levi2234 1 month ago Euler really was his day's chatGPT for mathematicians Reply Welch Labs · 1 reply @philippezevenberg1332 2 weeks ago We have ai but we dont have a ar app that can look at an equation and present it in 3d :/ 1 Reply @mythdream9833 1 month ago I'm still 14 but I kinda get it, I just need more knowledge about the basics like cos and sin. Reply @YMandarin 1 month ago (edited) complex logs can have multiple values? time for residues Reply @Sapientiaa 3 weeks ago What Python libraries are you using? Reply @worrierqueen5695 1 month ago The missing piece of the puzzle of course ... 🤣🤣🤣 Reply @Tletna 1 month ago Can you show that taking the complex conjugate results in the distance (magnitude) and that 2 to the bi having a magnitude at all even makes sense? I always thought making the complex plane coordinate grid was contrived in the first place and also e^i*pi = -1 was contrived as well. And, when I say "contrived", I don't mean right or wrong but simply an arbitrary definition. I mean, what is even the physical meaning behind 2 or e to the something times i power? Yes, you go over this in your video but I'm still not convinced on why Euler took the path he took or why we consider it 'correct'. Reply 1 reply @protektwar 1 month ago Mathematics and in SDR ;) Reply @gruntholo 1 month ago I came in here for grape juice Reply @willcool713 1 month ago So, this is what I don't get. Why is it that in physics, theta i is treated as a single dimension, summable between dimensions, and not a particular unique vector associated with each dimension. In a 3 dimensional world, I would expect there to be three complex planes, and therefore three unique complex directions, one associated with each axis/number line. Yet in physics, complex 3-space is three real dimensions and one complex component, when you're plotting EM waveforms, for instance. What gives? Reply @BalabhaskarSreelal-bp2er 1 month ago 0.693 is ln(2) Reply @JeffreyLopez-d1i 6 days ago Ebert Ridges Reply @kylewilliamrobertson5121 1 month ago Getting back to your roots 😆 Reply @virgiliovargas3052 1 month ago You're as smart as you're handsome Reply @ironfistgaming8945 1 month ago can you also please tell about the music used? Reply @VectorJW9260 12 days ago Hey, what is the music at 1:06? Reply @AstronutCymru 1 month ago Anyway to this in the UK? Reply @RyanWarm 2 weeks ago 95081 Howard Pines Reply @divermike8943 1 month ago I can imagine that the sin wave is the complex projection in the plane Normal to the cos projection. Is that correct? I wish that had been shown in 3D. Reply @Superstar-nl5tl 1 month ago please teach me math so I understand this video Reply @BlakeZachary-c6i 1 month ago Kessler Fords Reply @mimwarlick1604 1 month ago Is it possible to travel in the direction of the square of negative one…is it possible to travel in the i direction? Reply @ConsciousExpression 1 month ago (edited) e^(itau) = 1 is prettier Reply @Bender2k14 1 month ago Love your videos, and I especially love your music. Since you are preparing to put this content into a book, I have an improvement suggestion. At https://www.youtube.com/watch?v=f8CXG7dS-D0&t=531s, you say > Finally, it is very helpful sometimes to express complex numbers in polar form, where instead of writing the real and imaginary parts, we give the magnitude and the angle measured from the x axis. I have two issues with "x axis". First, there is no "x" here. You could say "horizontal axis", but you already (shortly before this) said the horizontal axis is the real axis, so I think it is better to use "real axis". Second, "angle measured from the horizontal axis" is ambiguous. From the positive half or the negative half? Clockwise or counterclockwise? Of course you mean counterclockwise from the positive real axis. It is ok in a video because your visuals resolve the ambiguity, but in a book, I think it would be better to include a longer phrase like this. 2 Reply 2 replies @donelson52 1 month ago What allows you to set a vertical axis imaginary? Reply @leonardomutti7525 1 month ago How do we know at 9.39 that the conjugate of 2^(bi) is 2^(-bi)? Reply @brian9438 1 month ago Seems legit. Reply @alternateash 1 month ago At 8:52 shouldn’t the magnitude of (3 + i) be square root of (3^2 + i ^2), so (9 - 1)^-2 = 8^-2? Instead of square root of 10? Reply 1 reply @ee2745 1 month ago Where do u find and buy old books? Reply @geoffresmart 1 month ago As a math student, i am so sick of math YouTubes constantly telling me how “elegant” equations are. Stop it Reply @mohammadwasilliterate8037 1 month ago Damn it am drunk I can't keep up. face-green-smilingface-green-smilingface-green-smilingface-green-smiling Reply @ManwithNoName-t1o 1 month ago starts at 24:40 1 Reply @MostlyIC 1 month ago Welch, your "zooming in and zooming out" description left me unimpressed, you're explaining the derivative at a point x but you're drawing a line to/from/(I can't tell exactly) to some other point x/2, or x/4, or ...., that's not how derivatives are defined, your explanation makes no sense to me. Reply @jamyjr235 1 month ago 08:50: what about third binomial formula? Reply @baglesac5806 1 month ago i guess your book is only available in the USA. Reply Welch Labs · 1 reply @SP-ny1fk 1 month ago They wrote in latin - that's interesting Reply @nickacca 1 month ago I'm not able to understand unfortunately this looks really interesting though Reply @ToolTechSoftware 1 month ago How do you order from Sweden? Reply @larrye.goinesjr.1535 1 month ago The Density Vector Value Is A Normal Of The Unit Sphere's Surface With The Center At The Origin, Any Values Not At A Normal, Are Not Valid Numbers! Reply @jb76489 1 month ago What’s the music playing at the beginning Reply @JamesaGray-b1l 1 month ago Ernser Island Reply @jonathanv.hoffmann3089 1 month ago 🎉🎉🎉 Reply @zitools 3 weeks ago Thanks man. new sub# 538K+ Reply @G11713 1 month ago 9:38 Could use some clarity. Reply @wieslaw.mlynarskiwiesaw6642 1 month ago 26.6+18.4=45.0 != 55 Reply @franzliszt3195 1 month ago Great. Please give free pdf link. Thank you Reply @zkzkzkzkz-z-z 3 weeks ago we sleeping to this one tonight 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥 💥v Reply @qa1e2r4 1 month ago I will go to the bank today and show them that by dividing my debt by a trillion I essentially owe them no interest on it as once I go to my localized area the interest is close to zero. Hence no interest was generated.... Brilliant no? Reply @VoodooD0g 1 month ago iam not even good at math, i dont know why i watched this Reply @saintperthnorthcloud3850 1 month ago I love the storytelling however math hates me to the core and I'm powerless in all equation. Reply @DrMikeE100 2 weeks ago You repeatedly make some small errors. For 2 to the power 5, the power 5 counts the number of appearances of the factor 2 - it does not count the number of multiplications, which is 4, not 5. You did that a few times... Also, each 2 is a factor in the multiplications - not a term. (Terms are added or subtracted, not multiplied.) 1 Reply @joaquinsolissilva6206 1 month ago Comon bro international shipping for the book 😢 Reply @alanday5255 1 month ago So what if the math is not wrong, but our understanding of the physics is still limited.. Reply @wmpowerwithin 1 month ago 8/20/224 e to the power pi = -1 Reply @somechrisguy 1 month ago Oiler Reply @KilgoreTroutAsf 1 month ago It is a formula, not an equation Reply @tikaanipippin 1 month ago Euler always looked "Pi i"ed in his portrait, it seems to me. Reply @michaldlugosz1965 12 days ago 25:55 what is the music name? Reply @weakw1ll 1 month ago 6:25 writing was the downfall of mathematics, if they had blender back then they shoulda just done visual graphs and intuitive visualizations instead of creating 300 different ways to yap and make math very convoluted smh!!! Facepalm 2 Reply @vectorsahel5420 1 month ago I have no idea whats going on Reply @joseopiyo9669 2 weeks ago 18.4+26.6=45😊 Reply @merfious3453 1 month ago 22:16 yan tuloy deadline na bukas Reply @Dunkle0steus 1 month ago it really sounds like you're pronouncing "calculus" as "kakulus" Reply @franklee3800 1 month ago the rainbow makes me turn the video off. Reply @deniedprosperity4144 1 month ago 4:45 Reply @nukiolbartes6279 1 month ago how does it sound? Reply @Dia.dromes 1 month ago Euler? Signalis reference? Reply @shawonhasnat7105 1 month ago ❤ Reply @imeprezime1285 1 month ago Beauty pageant in math😂. Sounds funny Reply @dreamycalculator 1 month ago lost me at 10:00 1 Reply 1 reply @valom3141 1 month ago what is the song plssss??? Reply @derekofbaltimore 1 month ago I still dont even understand why 2 to the 0 equals 1 I guess i could never understand the rest of the video then Seems interesting though 1 Reply 2 replies @nowymail 1 month ago i sin x? I went a step further. I sin xxx. Reply @noahnewman8264 4 days ago Anyone have the background music at the start? Reply @waqarahmed4091 1 month ago face-red-heart-shape Reply @kdeuler 1 month ago Great presentation. (And i’m no relation.😔) Reply Welch Labs · 1 reply @EyeIn_The_Sky 1 month ago if we have to make stuff up to preserve a previously made up property in regards to the 2 or any number to the power of Zero is one just goes to show our system is wrong but no one wants to talk about that, just sweep it under the carpet and keep on going down blind alleys ending up with ludicrousness and shit like string theory. Reply @mikithekynd 1 month ago 5:02 You completely lost me there. Up until that point you were describing everything step by step, but then you casually took the concept for "i" out of the blue without any explanation...and then began describing parts of it after you've already scrambled my brain 😅 which in turn made it tough to pay attention and learn. Reply @KOC6H2NO23 1 month ago Maybe my english is not good enough, but why do you call this identity "equation" ? Reply @cerosietebpositivo351 1 month ago As shown 1+2i angle is 63.43...😅 Reply @prakashraj4519 1 month ago Deja vu????? Reply @Nawdog 1 month ago 2^5 isn’t (2x2)(2x2x2). Arcfunctions? Reply 1 reply @dng88 1 month ago Why just ship to USA? Reply @WebsterWinston 5 days ago 28011 Medhurst Orchard Reply @pratikkute 1 month ago 🤯🤯 Reply @worldnotworld 1 month ago At 6:42, don't we have simply that b=0? Reply 2 replies @creamcheese3596 1 month ago MATHS not MATH! Reply @punk3900 1 month ago Like if you don't get it Reply @phibik 1 month ago Yo I have a video about 3d (actually 4d and 5d) complex function graph accros time, I think that the 3d colored graph you made has similarities with mine! 2 Reply @TheEndermanMob 3 weeks ago 20:29 is i/100 not 1/100 Reply @SampleroftheMultiverse 1 month ago 14:45 Reply @richard6381 1 month ago HOMER NO UNDERSTAND Reply @chudleyflusher7132 1 month ago I tried to explain this to my evangelical neighbor and he told all our neighbors that I worshipped Satan. I think we’ve reached peak science and mathematics and these people are going to drag us backwards. Reply @TimRobertsen 1 month ago Aaaaahhh! ... what :p Reply @onixotto 1 month ago I don't think so bro. Reply @dovaogedot 1 month ago What the fuck does x << 2 mean? Bitshift? Reply @SahilAnsari-sv1qg 1 month ago I am from India and I'm a very big fan of you. I'm a student that's why I don't have enough money to buy your book.Could you give some discount for students. I will very grateful.❤ 1 Reply @Pi-SquaredOrbitmath 2 weeks ago 11:50 Reply @fillstlauren7256 1 month ago Velocity Reply @TheSlimbee 1 month ago Too fast for me 😞 Reply @HoustonPaul 1 month ago Bro what this mean Reply @Blubpaule 1 month ago no shippin to EU. very sad -.- Reply @jonb4020 1 month ago "What could it possibly mean to raise 2 to the power of the square root of -1" you ask. And the answer is - who cares? Why not spend your time doing something useful for the world instead of wasting it with useless maths questions? Reply @SkinkUA 1 month ago cap! Reply @saganandroid4175 1 month ago You need to explain more terminology as you move along. And why are you in such a rush? Reply @ucngominh3354 1 month ago hi Reply @deckiedeckie 1 month ago Slide towards ignorance... Reply @GoGetFletch 1 month ago For an expansive mathematician, only shipping your book to the US is very narrow minded. I am very disappointed in you. Reply @баракобамаулетелвотпускнапараг 13 days ago 14:29 Reply @KalashnykAndrii 1 month ago 0:43 Петро Оліксійович, це шо зліва, ви? (Only Ukrainians will understand) Reply @ScottStokes-y2d 1 month ago What? Reply @Serghey_83 1 month ago Видел уже Reply @procactus9109 3 weeks ago I was going to watch this, but ruined straight up with imaginary numbers.. Its not imaginary if just calculate it. Reply @gtagamerm4169 1 month ago You did it which 3B1B couldn't Reply @amperin8999 1 month ago I am never gonna understand this video😢 Reply @yogirecords4726 1 month ago (edited) Another way of looking at is is like this: 2 to the power of 3 = 2x2x2 So you have the original 2 then you multiple 2 more times by 2 (ie 1 time less than the superscript symbol 3) 2 to the power of 2 you multiply by 2 only once because the power (superscript) is 2. 2 to the power 1 on you multiply no times so it remains as 2. 2 to the power zero is I logically as you cannot multiply less times than 0. It is like saying multiply 2 by 2, -1 times. That is impossible and so it’s meaningless to talk about it. The entire concept of multiplying a negative number by a negative number is a mind virus of nonsense. It is completely illogical and has no relation to reality at all. If I have a pool of water and I remove 1 m3 of water I could say I have -1m3 of water. If I multiplied the amount of water removed I could multiply it by 3 and I’d have 3 times less water so I’d have -3m3 of water. I cannot multiply the missing water by a negative number. That is meaningless. Reply @roberttelarket4934 1 month ago There is no way a novice can understand this since you talk a mile a minute! Reply @SOU70hgJXNROe9bNtuHF 1 month ago Too complicated. Reply @Emry11 1 month ago (edited) Interesting clip but you start from the first seconds with a false premise that is so rampant among academics and it is very sad and disappointing. Basically, there exist nothing called i=sqrt(-1) That is a non-existent definition and nowhere in math history any reliable source has ever defined it. The true and unique definition is i^2 = -1 and from that no one can possibly deduce i= sqrt(-1). To make this clear once for all, let's (for the heck of it) assume that i=sqrt(-1) exist (again, it doesn't but we pretend it does!). Then we will have: -1 = i^2 = i*i = sqrt(-1) * sqrt(-1) = sqrt( -1 * -1) = sqrt (+1) = +1 which wrongly implies -1 = +1 Besides, try the Euler's equation itself: e^(ix) = Cos(x) + i Sin(x) and try to substitute i with sqrt(-1) and see if you get the same result which of course you don't because i is the complex number and can't be substituted by any other definition. Another obvious example would be e^(i * Pi) = -1. Try to substitute i with sqrt(-1) in that and see if e^(sqrt(-1) * Pi) will be equal to -1 which you won't be able to show. I hope that you and all others are now clear about this very common and unfortunate mistake and actually mis-definition of the complex number "i" and won't use that false definition from now on. Good Luck! Reply @cparks1000000 1 month ago (edited) They're called "complex numbers" and not "imaginary numbers." It makes my ears hurt. Otherwise, the history is very interesting, while the math lacks a bit of nuance. 23:20 You missed "0" as an answer. Reply @themonkeydrunken 1 month ago This would be so much easier to follow without the intense vocal fry. Call a voice coach or something please Reply @pelasgeuspelasgeus4634 1 month ago At 8:06 you say that euler saved calculus. Nope. He completely destroyed it. It's also obvious that the cornerstone of imaginary numbers insanity is that x^0=1 which contradicts both the exponent definition and formal logic because how can a number be multiplied with itself zero times and give 1? But you need that to be 1 and not 0 as it's logically expected. Reply @migoo4469 11 days ago 58008 Reply @hg2. 1 month ago (edited) Why do you Millennials talk with this semi-sing-song voice that drops the volume on the last couple words of the sentence? You never hear professional narrators talk like that. [Could you work on not dropping the last couple words of the sentence? The "last word" may be obvious to you, drop with when you drop your voice volume to almost inaudible, it's too hard to hear. It's uncomfortable. It's like you don't want us to hear what you're saying.] If you just "can't help it", maybe you should get a professional narrator to read the text, or have MS Word read the text. Reply @Incalculable_Kyle 1 month ago (edited) the 2 equations shown at 9:15 are NOT equivalent. I took calculus and never saw the left equation in my life so I did some math and it turns out you fudged the numbers in the last step. You could have fudged the numbers somewhere else too but honestly I can't be fucked to check it. For reference, the actual numbers in the last step are 7.07 ∠55° = 4.06 + 5.79 i usage of basic SOHCAHTOA to prove it Cos ∠55° = x/7.07 7.07 cos ∠55° = x 4.06 = x Sin ∠55° = y/7.07 7.07 Sin ∠55° = y 5.79 = y Also, I know its intentional bcuz in the graphic shown, the 55° angle is NOT a 55° angle. You have intentionally shown a misleading graphic to make your fake numbers look more believable. Highly disappointed in you as a creator 1 Reply 1 reply @drawing_guy_guy 2 weeks ago why do you sound like you have a voice crack every minute? Reply @terry123 1 month ago You explain too fast Reply @GeorgeKlinger 1 month ago Kamala ads?! Really? 1 Reply 1 reply @riccardorodia6312 1 month ago √-1 is not i. i is one of the two solutions to the equation x^2=-1 Reply 1 reply @user-Natarito 1 month ago I don’t want to seem like a dumbass but, what? Reply 1 reply @ali_barry 1 month ago Try to graph riemann zeta function in 3d Reply @cambrierogers9449 1 month ago can you like a video twice? Reply @kamurashev 1 month ago 🫠 Reply @Kyoz 1 month ago 🤍 Reply @geoffresmart 1 month ago Also… this was NOT explained visually. This was explained verbally, and used graphs and equations to occasionally provide a triangle. If this is explained visually, then so is every math book ever. Reply @arktessellator_10 1 month ago Still not a satisfying explaination 😂😢 Reply @Arivarul 1 month ago ❤❤❤ I am the first to notice 😊 1 Reply @larrye.goinesjr.1535 1 month ago Imaginary Numbers Are Values That Have Gone Past Their Usefulness. A 45 Degree Staircase Is Only Useful Until You Get To Your Desired Floor. A 45 Degree Staircase Doesn't Go All The Way To Heaven Because It Is Largely Impossible?!? But A 0 Degree Staircase Is Simply Useless?!? Keep Your Input Data Coordinated With Your Output Data!! Reply @PersonManManManMan 1 month ago (484# Reply @joshuawhitworth6456 1 month ago Boring. Reply @Wonders_of_Reality 1 month ago Sorry, I closed the window at the end. When I see tattoos, I immediately feel sick. Yeah, mathematics is beautiful, but not always. Reply @kensmith5694 1 month ago It really is a funny joke he is telling in complete dead-pan Kamala got it. You can see her expression. Reply @yisahak 1 month ago First view Reply @michaelqiu8687 1 month a