Euler’s real identity NOT e to the i pi = -1

Euler’s real identity NOT e to the i pi = -1

Welcome to another Mathologer video.
Everybody who watches these videos knows Euler’s identity e to the I pi is equal
to minus one except it’s not really Euler’s identity. The mathematician Roger
Cotes already wrote about it in 1714 when Euler was only seven years old. I
actually find it a bit sad that people associate the math super hero Euler with
a result that’s not really by him rather than with one of the zillion’s of his
really original amazing discoveries. Of course it’s really sad for Roger Cotes
since he doesn’t get mentioned for anything and nobody’s ever heard of him.
Anyway I thought I make a video about the real Euler identity, the identity that
made Euler into a mathematical celebrity. It’s this one here. Very curious, right? It
says that pi squared over six is equal to the infinite sum of the reciprocals
of the squares. This identity is a very surprising answer to the problem of whether the infinite sum on the right adds up to anything nice. A lot of the most famous
mathematicians of Euler’s time had tried and failed to answer this problem and so
it was a bit of a sensation when in 1734 Euler managed to find the answer. And of
course that the answer involves pi with no circle in sight anywhere added really to
the appeal of this solution. The result is not only pretty but also turns
out to be important. Up there is the definition of the famous Riemann zeta
function which is the function at the heart of the Riemann hypothesis, one of
the most important unsolved problems in mathematics. So what Euler managed to do was to find the precise value of the Riemann zeta
function at 2. Not only that, he also did so for all even numbers. Now it turns out
that all these values involved the circle number pi. For example, this up there is Zeta evaluated at 4 and this is Zeta evaluated at 6. To this day nobody knows whether zeta evaluates to anything nice at the odd
values 3, 5, 7, etc. Now Euler was also interested in assigning
reasonable values to the exploding sums that you get when you try to evaluate
Zeta at negative values: 1+2+3+… you know that one, right? In
fact, he was one of the first, if not the first mathematician to really
demonstrate that there is a lot more to these sums than meets the eye at first glance.
If you dare check out my video on this really mind-boggling topic. Anyway my
mission today is to motivate and animate Euler’s proof of his wonderful
identities for you. So here we go. To start with it’s not clear at all that
this infinite sum will actually add to a finite value and if it does add to a
finite value whether it adds to anything nice. If you’ve already looked a bit
at infinite sums you may have come across a beautiful closely related
infinite sum. It gives a lot of insight in this respect so let’s warm up with
that infinite sum. To get to it we first rewrite Euler’s sum like this.
So, for example, we write 1 over 2 squared as 1/2 times 1/2. Make a copy and let’s
shift the blue bits one to the right. What results is this other infinite sum which
is also very pretty. Let’s compare the two sums term by term. Okay, in the green
we have 1 on top and at the bottom. Here the green factors are the same and the
bottom blue one is bigger … green the same, bottom blue bigger, green the same bottom blue bigger and so on overall it’s clear that the bottom sum is bigger than
Euler’s sum at the top. Now for the really ingenious argument that evaluates the sum
at the bottom. Let’s chop off this infinite sum at the fourth term there
and calculate this partial sum. Now get ready for a bit of mathematical magic. Here 1 times 1/2 is equal to 1 minus 1/2, neat right?
1/2 times 1/3 is equal to 1/2 minus 1/3, and so on, now we get rid of the brackets and
this gives you a strange mathematical telescope of the pirate variety 🙂 Let me
explain. This here is one of the segments of the telescope and here’s another one.
Okay now let’s collapse the telescope: – 1/2 plus 1/2 is 0, – 1/3 plus
1/3 is 0, and so our partial sum evaluates to 2 minus 1/4.
Now we chopped off the infinite sum at this term over there. If instead we would
have chopped off at that term here a longer telescope would have given us the
partial sum 2 minus 100. Now the further down the sum we cut, the smaller the
number we minus gets. At infinity it vanishes and so our telescoping sum adds up to 2,
exactly. Okay so these infinite sums can really add to nice values and so Euler
& Co. had some hope that the same might be the case for the closly related
infinite sum they were chasing. Also since the sum is greater than the Euler
sum we know the Euler sum adds to a finite number which is less than 2. And,
in fact, pi square divided by 6 is approximately 1.644 which is less
than 2. So it’s all looking really good. Ok warmup is over. Now how did Euler
manage to prove his identity. Well he starts with the so called Maclaurin
series of the sine function. Most of you who’ve done some calculus are familiar
with writing sine as this sort of infinite polynomial. Definitely Euler and
all his contemporaries knew this very well. Now to present a complete picture
and to motivate where Euler’s idea for his proof comes from, it’s important to
know how mathematicians derive this formula and I’ll explain it in a moment
but for now let’s have a look at this identity. Obviously, the sine function is
derived from the circle and it should therefore not come as a surprise that pi
is hiding in sine in a couple of different ways. For example, setting X
equal to pi sine becomes 0 and so we get this funky expression for pi here which
is almost as nice as Euler’s identity, right? Question for you: How could you use
this expression to calculate better and better approximations for pi? If you’ve
got an idea tell us in the comments. All right, let’s say you suspect sine of x
can be expressed as an infinite polynomial like this. How do you go about
finding those coefficients? Well a is easy. Just set x equal to 0 and sine is
equal to 0 and so is this expression over there and this means that a has to
be equal to 0. Now let’s find the derivatives on both sides of the
equation. If you don’t know what a derivative is just go with the flow. Here
the derivative of sine X is cosine X, the derivative of a is 0, the derivative of b x is b, the derivative of c x squared is 2 c x, the derivative of d x cubed is 3 d x
squared, etc. Ok, now if we set x equal to 0 again. This gives on the left side
1 and here on the right side 0. And so b is equal to 1. Now just rinse and
repeat, find the derivative, set x equal to 0 and conclude that c is equal to 0.
And again and again and again and again and that gives you all the coefficients.
So if sine can be written as an infinite polynomial that’s what it’s going to be.
Anyway that’s where the Maclaurin series for sine comes from. This identity
actually works for all x but that it really does of course requires a
separate proof. Let me just demonstrate how well this
works by showing you how the partial sums of this infinite sum are better and
better approximations to sine of x. There we go. Let’s plot sine of x, there we go and first partial sum that’s just a line, second partial sum is this
cubic here, and then we keep on going, you can see these guys wrap closer and
closer to sine and at infinity they coincide. Anyway, as you’ve seen, the
infinite sum on the right is constructed term-by-term based on higher and higher
derivatives of sine at zero. That’s really wonderful and actually gives similar
formulas for all the really famous functions, like the exponential function
or cosine, etc. At the same time this method of pinning down this
infinite polynomial is different from what most of us would try first for
finite polynomials, right? For example, for a linear function like this we
construct it maybe from two points on its graph. Or for a quadratic function from
three different points. And so it’s kind of natural to see whether we can also
determine that infinite right side up there by taking a similar approach. And
that’s exactly what Euler does. What he does is he constructs the infinite
polynomial in a different way from these special points here which, of course,
correspond to the zeros of sine. Okay let’s first focus on the middle three
zeros. That part of sine looks like a cubic curve and we can straight away
write down a cubic polynomial that has those three zeros. Here it is. Pretty obvious
it’s got these three zeros. Let’s graph it. Well it does have those three zeros
but otherwise it’s not a great fit. Of course this is just one of the
infinitely many cubic polynomials that has these zeros. We get the other ones by
putting a constant in front and varying it. So let’s just do that. Okay down, down
down, down, back up a bit and so we get a very close fit when the slopes of sine
and the cubic coincide at zero. And it’s not hard to see that this
happens when the constant is 1/pi squared. Well let’s just run with it. Let’s
clean up a bit. first pull those constants into the brackets. Two 1s there
and there, put the factors in order and combine the last two factors as you
learned in primary school. Hope you paid attention. Okay looks neat and now it’s
also easy to check that this cubic just like sine of x has slope 1 at 0. Next
let’s repeat the same calculation for the middle 5 zeros. The polynomial we get
here is of order 5 and it looks like this. In terms of the algebra you get
exactly the same answer as before except you get another factor that features 2pi instead of a pi and which takes care of the outer zeros 2pi and -2pi.
Now just repeat over and over adding two zeros at a time. You can see a closer and
closer fit and at infinity we’ve got coincidence again. And so Euler now has
two infinite polynomials that are both equal to sine x, right? Now since the
infinite expressions are equal in a way, by expanding the infinite product we
expect to recover the infinite sum. Let’s just have a look at how this pans
out numerically. First for the infinite sum. So that’s what it is and here the
smallest terms you get when you expand the first couple of factors of the
infinite product. As you can see, you get pretty good coincidence in the
coefficients which gets better and better the more factors you use. So this really
seems to work. Now how do you expand the whole infinite product? Well, just start
expanding one factor at a time. So highlight the first two factors and
multiply them together and you get this. Take the result and the next factor,
multiply and you get this guy. Now let’s also highlight the x cubed term
because that’s going to be very important in a second, and focus on how it
evolves. Okay, next factor is there. Multiply and you get this one here, and
again. And now just have a look at the green. It’s pretty clear what’s going to
happen in the end once you’ve finished expanding. The x
cubed term will look like this. But remember this x cubed term of the
expanded infinite product has to be equal to the x cubed term of the
infinite sum and so we get this equation here which simplifies. First multiply by minus 1 then times pi squared and there you have it, Euler’s real identity, pretty ingenious
argument isn’t it. And all of other identities that I mentioned earlier,
like this one up there, you also get by comparing coefficients
of the infinite sum and the expanded product. For example, you get the identity
up there by comparing the coefficients of x to the power of 5 in the Maclaurin
series to the one in the expanded infinite product. Maybe give this a shot yourself.
It’s really very rewarding if you managed to do this on your own. If you
get stuck I’ll also do the nifty calculation in a separate video on
Mathologer 2. Eventually of course there’s a lot more I could say about
this identity. Most importantly I should stress that what I’ve shown you is not
a complete proof. It still takes a bit of work to make this argument completely
rigorous but it can be done, However, what I showed you is pretty much all that
Euler had when he decided to go public with his result. In fact, believe it
or not :)he didn’t have access to Mathematica and so all the numerical
evidence to support his argument he had to produce by hand and he actually had a
lot less numerical evidence than I showed you. Now it still took Euler a
couple of years after he published his identity to come up with a completely
rigorous proof. Now to finish let me just mention two more amazing facts about
Euler’s sum. For my first amazing fact, let’s have another look at Euler’s way of
writing sine as an infinite product up there. If we set x equal to pi
divided by 2, sine becomes 1 and when we write the factors on the right
as fraction, we get this. Solving for pi divided by 2 gives the so called Wallis
product, named after the mathematician John Walli, a really beautiful and
useful infinite product featuring squares of all the even numbers in the
numerator and squares of all the odd numbers in the denominator, really really
nice in it. Amazing fact number two: a lot of you will be familiar with the Leibniz
formula, this absolutely wonderful identity up there. I didn’t know about
this myself until recently but Euler actually points out in one of his papers
that this identity can be derived in exactly the same way as Euler’s sum, the
sum that this video is about, by writing the function 1 – sine x as a product
using its zeros so there’s pretty much all you need and then comparing the
x coefficient to the Maclaurin series with that of the expanded infinite
product not hard once you know so maybe one of you may also be able to supply
the details in the comments. Oh, by the way, although this amazing identity is
named after Leibniz, it, as well as the Maclaurin series of all the main
trigonometric functions were already known to Indian mathematicians at least
300 years before Leibniz and Maclaurin were born and it may very well be due to the mathematician Madhava of Sanggammagrammar. And hardly anybody knows about
this or cares how very say it anyway knows about this or cares. How very sad. And this is it for today. As usual let me know how well this video work for you and see you next time.

100 thoughts on “Euler’s real identity NOT e to the i pi = -1

  1. Thanks for reference to Indian Mathematician "Madhava of Sangamagrama" in the end. We definitely care.
    The series is also called as Madhava–Leibniz series.

  2. The only thing I learnt from this video is the identity e^(iπ)=-1, which is fundamental in maths. All the other identities just exist.

  3. Thanks for your great videos. Maybe you should also make a video about how Indian mathematician derived such formulas?

  4. Насчёт математики сильно сомневаюсь, но английский на этом канале я выучу.

  5. You can find a good exposition of Euler's development on this and other subjects on the book 'Euler, the master of us all', by William Dunham. The author adds comments which really help understand how they used and thought of math; It's really worth 🙂

  6. The resultant mechanical force of gravity forming an i force square root of -1 operating in Einstein's equation of variable mass activated by two vector plane of horizontal and verticality in between the energy transfer between perpendicular vectors of spiralities that forms a gravity force as operated in between inductive spiralities of variations and hence it was important to introduce square root of -1 as interactive link in between magnetic and electric field as a locus of sliding ingredients of gravity of middle point of gravity forming an erratic by its circular pi shifted for an elliptical orbital variations.
    Sankaravelayudhan Nandakumar.

  7. Hi Mathologer,

    Most of the proofs relating to pi tend to involve the sin/cos functions. My question is: why is the 'natural' sin graph in base pi?
    It obviously makes sense because numerically the Taylor Series appears to converge to a root at pi, 2 pi etc, but how can we be certain?

    The derivation from unit circles to the sin graph doesn't quite explain to me why pi radians is the first root. The x-axis looks like it could be to any scale.

    Your videos are awesome, please keep making them, especially number theory ones.



  8. Take natural log on both sides. Like 1*2*3*4… 2 ln. So 2 phi and inverse phi square by six. Whenever you see a spiral use e power or natural log. Even for imaginary numbers. A triple right angle curves.

  9. Sorry, but the sentence "at infinity they coincide" does not have any sense, there is no infinity in this context.

  10. lol no one has heard of him probably because some Brit made up that he discovered it. Living beside the British they have a tendency to make up that a British invented something and told nobody, it gets kind of boring to be honest. like the germans inventing the jet but they stole it from whittle and told nobody or they invented blitzkrieg and told nobody. Or they invented gun powder and told nobody. revisionist history is quite popular in the uk.

  11. I think you have to have a closed mind to say that ζ(3) and ζ(5) etc aren't "nice" numbers just because they aren't apparently related to e or π. From an objective point of view they're just as special or (to use a non-mathematical term) "nice" as their more familiar e and π counterparts.

  12. "… as you learned in primary school. Hope you paid attention. Heh heh."

    Aloha Mathologer. I love your videos. But do you have any videos for those of us that didn't pay attention in primary school? 🙂

  13. You know what's weird with pi all this time? It's irrational in the sense that it's not a ratio of two integers, but it's practical in the sense that it's a ratio between the circumference and the diameter, two lines that can be drawned or at least imagined in the real world.

  14. What is called Euler's identity is e^(pi i) + 1 = 0, not e^(pi i) = -1. The last formula, say Cotes' identity, lacks the essential aesthetics (see Wikipedia about Euler's identity), so Euler's identity is not Cotes' identity.

  15. The infinite series for sine of x was actually discovered by Madhava of Sangamagrama from India 🙁
    Not any european mathematician.

  16. how does it work for me?… i look it 3 times, and the third i took a sheet to calculate. great approach to understand what Euler had really done. Thank you to talk about Roger Cotes.

  17. How do you know the Indians aren't lying? Why would they not have come forward at that time hundred of years ago, to participate?

  18. Mathologer and 3Blue1Brown are honestly legends, revolutionaries. You guys change the world with every video. Absolutely amazing communicators, a skill sadly rare in higher education and complex topics. decades from now, you guys will be like the Feynman of math education. Keep up the amazing work

  19. Wait, can you say that the coefficients of x^3 are the same? That’s like saying 3x^3+4x=2x^3+9x this 3=2, right? What am I missing?

  20. Hi, great video. Love your videos but only watching on and off. I am more of a data base person and programmer. Not sure you still make videos, but if you do then show us why we still cannot determine i. I am sure people have pointed this out before, with e to the i pi equals -1, you only have one unknown and this should be easy to calculate. Yeah I know! If you have made a video about this can you send me the link please. Thanks

  21. Nice video, but should've mentionned that, AFAIK, the only result we have regarding ζ(2n+1), n >= 1, is due to Roger Apéry the french mathematician who (unexpectedly) proved in 1978 that ζ(3) is irrational. Actually it's a bit more complicated than that see here:'s_theorem
    About the proof Van Der Poorten said: Apéry's incredible proof appears to be a mixture of miracles and mysteries
    Actually there is another result due to Wadim Zudilin back in 2001: at least one of the 4 values ζ(5), ζ(7), ζ(9) and ζ(11) is irrational.

    Sources :

  22. Nice video, but should've mentionned that, AFAIK, the only result we have regarding ζ(2n+1), n >= 1, is due to Roger Apéry the french mathematician who (unexpectedly) proved in 1978 that ζ(3) is irrational. Actually it's a bit more complicated than that see here:'s_theorem
    About the proof Van Der Poorten said: Apéry's incredible proof appears to be a mixture of miracles and mysteries
    Actually there is another result due to Wadim Zudilin back in 2001: at least one of the 4 values ζ(5), ζ(7), ζ(9) and ζ(11) is irrational.

    Sources :

  23. Nice video, but should've mentionned that, AFAIK, the only result we have regarding ζ(2n+1), n >= 1, is due to Roger Apéry the french mathematician who (unexpectedly) proved in 1978 that ζ(3) is irrational. Actually it's a bit more complicated than that see here:'s_theorem
    About the proof Van Der Poorten said: Apéry's incredible proof appears to be a mixture of miracles and mysteries
    Actually there is another result due to Wadim Zudilin back in 2001: at least one of the 4 values ζ(5), ζ(7), ζ(9) and ζ(11) is irrational.

    Sources :

  24. 0:22 I think he was the one who gave the cotes quadrature formula, a numerical method for finding the definite integral of a polynomial whose sample data of finite output values over equally separated input values are known beforehand.

  25. So what you're saying is that Euler recieved credit for something that another mathematician discovered BEFORE him? That's amazing.

    Normally, a theorem/law in modern mathematics has to be credited to the second person who discovered it because the first is usually Euler.

  26. Something as simple as x_i+1 = 2x_i – x_i^3/3 + x_i^5/5 – … = x_i + sin(x_i) should iterate to pi given x_0 is close enough to pi to begin with, say x_0 = 3.

    Since sin(x) -> 0 as x -> pi, x + sin(x) -> pi as x-> pi. This is only possible since the absolute gradient of x + sin(x) near pi is less than 1 (in fact close to 0), so modifications to the iterate will get smaller and smaller hence convergence.

  27. The Indian mathematician "Madhava" Mathologer refer in the end:

    1. "Calculating Pi, Madhava Style" –

    2. "9. Computing π – The Indian Method" –

  28. In the derivation of the Leibniz formula I think it should be: (1-2x/(1π))² (1+2x/(3π))² (1-2x/(5π))² (1+2x/(7π))² …

  29. Roger Cotes has been mentioned as either one of Newton’s teachers or his predecessors. Corrections (?) from Wikipedia:

  30. I'm struggling with the product formula for 1-sin(x):

    How did you find (1 – 2 x/Pi)^2 (1 + 2 x/(3 Pi))^2 (1 – 2 x/(5 Pi))^2 (1 + 2 x/(9 Pi))^2…? Especially the squares seem be coming from nowhere. Thanks for the great video.

  31. i am an indian and the mathematician you named had given a samskrit shloka to describe value of pi to construct perfect 'hawan kunds'

  32. Great video. It would be nice to have a video on early Indian mathematicians. Ramanujan is often presented as a boy wonder, sui generis . But he came from a long tradition. Perhaps team up with a historian of mathematics.

  33. I honestly had no real idea how much mathematics is discovered simply by seeking out syntactic patterns – literally notational patterns – the provable uniformity of which, or provable uniform variation of which, then supply the basis for strict (mathematically) inductive conclusions, or, more startlingly, the the straightforward bulk-handling of infinite quantities.

  34. A hundred years or so equalizes everyone, and no (man)person is an independent island at any time..? Even Pi-rates become x-potential? (Bad Math pun)

    This lecture is a good demo of how and why the maths of "Everything happens all at once", time duration timing = differentiation,
    has roots of probability 1,0 in Principle,
    the functional omnidirectional-dimensional Origin of Superspin Modulation Conception (=> Totality, correspondence),
    positioning by all-wave-package possibilities = resonance -entanglement sync,
    in the Eternity-now Superspin Singularity = integration -interval.

    Grand Unification Observation.
    But there's another classification of probability in potential possibilities dominance as it relates to the functions of i-reflection, closed total internal reflection of omnidirectional-dimensional substantiation Principle In-form-ation formulae in Mathematical Abstractions, to assemble/associate with the above video. (Not enough room in the margin(-al value) to comment..)

  35. 16:20 I'd always thought that coefficient at the simplest term to represent the freedom in space, like viscosity. What you're stating about the Indian mathematicians seems to draw the conclusion that some really cool man made stuff was created with the Euler like the Ellora Caves. Thanks-

  36. The Arabs and the Europeans cared about the Madhava series sums, 300 years before Leibniz, but the Indians doesn't care. How intelligent the Indians are?

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