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For this Christmas video the Mathologer sets out to explain Euler’s identity e to the pi i = -1, the most beautiful identity in math to our clueless friend Homer Simpson. Very challenging to get this right since Homer knows close to no math!
Here are a couple of other nice videos on Euler’s identity that you may want to check out:
(one of our Math in the Simpsons videos)
(by 3Blue1Brown)
And for those of you who enjoy some mathematical challenges here is your homework assignment on Euler’s identity:
1. How much money does Homer have after Pi years if interest is compounded continuously?
2. How much money does Homer have after an imaginary Pi number of years?
3. As we’ve seen when you let m go to infinity the function (1+x/m)^m turns into the exponential function. In fact, it turns into the infinite series expansion of the exponential function that we used in our previous video. Can you explain why?
4. Can you explain the e to pi i paradox that we’ve captured in this video on Mathologer 2:
If you own Mathematica you can play with this Mathematica notebook that I put together for this video
Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.
Merry Christmas!
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excellent explanation! it might be due to my own inabilities as a mathematician but I’m still confused on exponentiating a complex number. I understand that e^ipi corresponds with going around the complex unit circle, but if you gave me any other number, say, 5, I still feel like I dont know how to approach exponentials like 5^i
If you think e^(iπ) = -1 is beautiful, which it is, rearrange the identity to find out what π is equal to. I promise it will blow your mind. I get
π = -i ln(-1). I could be wrong, but I know I'm close. What do you get? Whatever you get, it will be mind-blowing.
3.4m views!
I prefer Euler's formula in the form e^(i * π) + 1 = 0, because 0 and 1 are fundamental constants derived from the axioms, whereas (−1) is essentially already an arithmetic operation in itself
Thanks to Homer now I've understood the meaning of Euler's formula.
That was the best explanation I've ever seen about this. Wow. THANK YOU! (I don't understand the 2k dislikes…)
Круто, но последняя часть с треугольниками совсем не обоснована. Почему так работает умножение комплексных чисел, я ещё могу посчитать, но почему в пределе получается полуокружность?
Wait what I don’t understand. 5:17 ((1+(1/n))^n)^pi=(1+(1/n))^(n*pi)
8:29 why is the first part positive when m = 1, but negative when m > 1?
Who you calling dummy dummy pi are not square pie is round 😅
I have always wondered, that since e is the limit of (1+1/n)^n, then for all finite n, this is actually a rational number so "when" does it actually become itrrational ?
Thanks!
Absolutely brilliant video.
Around 11:00, this is how (essentially) Richard Feynman explains complex multiplication.
Of all the videos I have watched about Euler's identity, none explains it better than this!
Explained most perfectly and in most excellently lucid manner!
If you have the background of studying how log with base 2 evolved with slope 0.693 near 0, this video makes you understand e raised to πi like a child's play! 🙂
15:30 if this is the easy way to explain it, i don't want to see the hard way
Fantastic!
At approximately 8:30 he encodes a negative number because i is square root of negative 1. That is the problem. When mathematicians use i they propagate the error all through mathematics. That’s why you see 1/12 recurring in physics and not -1/12 unless they use math with a proof reliant on the negative in i. Again, that encodes direction. An article of clothing in the laundry is the same inside out. But what an incredible human being to produce his lovely mathologer videos! I am a huge fan!!
Very cool demonstration!
Could have shown that you can multiply this pi to do a turn for whatever angle in radians you want by the end of the video
Did he glitch earlier in the video when he said i²=1, only to say i²=-1 later in the video?
A marvelous video! I've wondered about Euler's equation for years. The Mathologer works his magic by giving an explanation that even I can understand. Now I get it. Thanks, Mathologer!
beautiful
I have just now realised that complex plane is not the physical geometric plane. We just use this imaginary looking plane to look at a number visually. If one looks at the vector, it’s Pythagoras Theorem such that sqrt (x^2 + y^2) = radius = sqrt[[cos()]^2 + [sin()]^2] which is different to Euler's identity. Additionally, in a physical geometric plane, if you traveled 1 meter north and multiply by itself (which just means once), it doesn’t give 1 meter west.
Where as a complex plane is this e^i()=cos() + i sin() thing, in its simplest form. Although I can’t think of anything in exact form, but the closest thing I can think of physically is Potential and Kinetic energy in spring or pendulum (although it only works in the first quadrant because energy doesn’t have negative values). So if a physical unit has two states, and the two states are out of phase by 90 degrees, it kind of fits the complex number model very well, so in this case one can treat potential energy like an ‘i’ component of a kinetic energy and x*e^i() = total energy in the system and x*e^i() = KE + PE = x*cos() + x*i sin(), where ‘e’ is just like a fancy letter for this imaginary radius of the value 1 on the complex plane as this imaginary angle changes, since e^i(0) = 1 on the real number line, and that’s the only time it intersects the positive part of the real number line where things makes sense and brain can more easily perceive the reality, otherwise imaginary angle, imaginary radius or imaginary plane are not geometric. (But it’s so super interesting nevertheless).
That was so beatiful 🥹
Almost ten years have past and he has still not made a video about e^pi.
omfg. I finally understand e. I'm 43. I understood what e was before… but never understood why e was. Now I understand why e is. lol
love ur t shirt
Thanks that was so intuitive and clear ❤
I think it's useful with equations like this to show what happens when you slide the values in different ways, so in that sense (and many others), this is great. I still don't quite see how you're allowed to scale the triangles when stacking them up. Strangely it seems fine when you're just rotating around the circle but when they get bigger each iteration, seems off (even though I know it's legit).
What if π^{ie} ? 😂
There is just one problem, n*pi*i is not infinitely large…when n is infinitely large…
I like how the bank of Springfield is called BS.
Homer: 100% interest? WOO HOO!
pi is not accurate
Are not all numbers imaginary? Even real numbers are imaginary. It´s not that i can find "1" and "2" when walking in the forest, or on the shelves of a shop. I have to think "1" and "2". I´m just thinking about the names.. and the complexplane.. The complex- plane? Also, have you actually thought about what energy is? Energy is mathematical relations between physical phenomena.