Amir> I am trying to see if someone can help us understand it.

Well, there is the isomorphism between the rotation group in two dimensions, SO(2), and the group of unitary 1x1 matrices, U(1), also known as the multiplicative group of complex numbers of absolute value 1. But that, at best, is again a mathematical explanation (which I used in the interpretation above). I am sure there are many other interesting ways to look at this, but for the moment I cannot think of any.

Huh!?? You want a non-mathematical explanation of an equation packed with mathematical concepts and symbols? That is impossible.

The equation says that by turning around 180o (PI radians), you will be looking exactly backwards. (That was not an explanation, that was an interpretation.)

Prof. Olaussen,

My fascination isn't with the mathematical proof but why it works? We teach students that any positive number raised to any power will produce a positive number of zero and yet this equation shows that raising e to the power of i*PI yields -1! Amir

Amir> We teach students that any positive number raised to any power will produce a positive number

Well, you should stop teaching that! At least without the qualification that this is a property of real (in the mathematical sense) numbers. Or, more fundamentally, that the square of number is always positive.

Of course, how much fuzz one should make of the qualifications depends one the class being taught to...

Note added: Actually, the Euler identity may be used to "prove" many impossible identities, by using the fact that the inverse of the exponential function is multivalued:

eiπ = -1 ⇒ e2iπ = 1

Hence, for any number z, 1 = 1z = e2iπz. So, for any number z=-ix, with x real, we have found that e2πx = 1 :=D :=D

I agree with you regarding the level of the class being taught. We know the SQUARE of the SQRT(-1) is equal to -1. We also know SQRT (-1) * SQRT (-1) = -1 and not SQRT (-1 * -1) = 1. This is not about mathematical tricks and order of operations. It about an amazing identity that represents the basis for solving many practical problems in structural dynamics, seepage, etc. I am just curious if there is an explanation for why it works other than the Euler mathematical proof. Thanks.

This is from the web!

Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth".

So I am trying to see if someone can help us understand it. Amir

Amir> I am trying to see if someone can help us understand it.

Well, there is the isomorphism between the rotation group in two dimensions, SO(2), and the group of unitary 1x1 matrices, U(1), also known as the multiplicative group of complex numbers of absolute value 1. But that, at best, is again a mathematical explanation (which I used in the interpretation above). I am sure there are many other interesting ways to look at this, but for the moment I cannot think of any.

Kåre - I appreciate your comments and suggestion regarding the unitary 1x1 matrices. I hope we can find another way. Amir

The Euler identity (however pretty) is, in my view, of rather modest practical use compared to the Euler equation,

eiπx = cos(πx) + i sin(πx),

which I have already alluded to. However, that equation is again just a footnote consequence example of the Cauchy-Riemann equations defining the analytic functions of Complex analysis.

Agree and that is why I mentioned Euler.

Kare,

On a separate issue, are you familiar with matrices other than the identity matrix that are equal to their own inverse. Also, matrices other than the zero matrix that are equal to themselves when raised to any power? I would like to know the mathmenatical basis for such square matrices and if there are any practical uses. Thank you. Amir

Dear Amir, I appreciate that you want to give your students more than just a "dry knowledge".

I find useful https://en.wikipedia.org/wiki/Circular_polarization giving an explanation how-where-why Euler´s formula works. Pitty there is no description for the second figure - I agree that some students cannot understand without help that the green-blue-red curves are that of cos x, sin x and eix , and the first narrow goes from π on x-axis to the corresponding point on the red graph ... to the value -1.

@Kåre I like a lot your "The equation says that by turning around 180o (PI radians), you will be looking exactly backwards" - it is excatly what I see in the formula :)

Dear Viera,

Euler used Taylor series expansion of the sin x and cos x to derive his famous identity. This is still a mathematical explanation. Amir

Dear Amir, you´re right.

But I mentionned the circular polarization. For me, it is a different view.

As for matrices: involutory matrix A=A-1, idempotent matrix A=A2.

Dear Viera,

are there any practical applications for these matrices?

amir

Thanks Peter. excellent mathematical explanation. Amir

Amir Euler's identity is a special case of Euler's formula lets recall from complex analysis that states that for any real number x, e^{ix}=cos x+i sin x

where the inputs of the trigonometric functions sine and cosine are given in radians.In particular, when x = π, or one half-turn (180°) around a circle:

e i π = cos ⁡ π + i sin ⁡ π . {e^{i\pi }= cos pi +i sin pi .}

Since

cos ⁡ π = − 1 {cos pi =-1}

and

sin ⁡ π = 0 , {sin pi =0,}

it follows that

e i π = − 1 + 0 i ,

which yields Euler's identity: e i π + 1 = 0. Which proves the existence of God

Obviously, the math is correct.

What do you mean exactly by: " Can anyone explain it rationally and not mathematically?"

Mathematical explanations can go from the informal (like drawings) to the precise proof in a formal axiomatic system. What would be the difference between a mathematical explanation and a "rational" explanation?

Besides the Taylor series expansion of the sin x and cos x that Euler used to develop his famous identity, I am searching for some logic that beautifully connects the real to the imaginary. I have used complex variables to solve seepage problems, structural dynamic problems, ordinary and differrntial equations, and used conformal mapping but always go back to the following:

Benjamin Pierce, a noted American 19-th century philosopher, mathematician, and professor at Harvard University said of this identity “we can not understand it, and we don’t know what it means, but we proved it to be true.

Can you help Prof. Pierce understand it and its meaning? Someone out there may know how to provide an explantation and it isn’t Prof. Pierce

Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience. Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge." FROM the INTERNET!

Amir> matrices other than the identity matrix that are equal to their own inverse.

If M is such a matrix, it can always be brought to Jordan normal form by a similarity transform, M = S J S-1. The requirement that M2 = I (unit matrix) implies that J must actually be a diagonal matrix with entries ±1. So, the given form of M, with such a J, and S any SL(n, C) matrix, will be a solution. A parametrisation of all possible solutions, complete but not over-complete, is a different and more difficult problem.

Amir> matrices other than the zero matrix that are equal to themselves when raised to any power

You may apply a similar method to analyse the requirements that Mn = M for some positive integer n (or for all positive integers n).

Thank you Kåre. Your thoughtful response is very much appreciated.

You said:

"Benjamin Pierce, a noted American 19-th century philosopher, mathematician, and professor at Harvard University said of this identity “we can not understand it, and we don’t know what it means, but we proved it to be true."

The same remark could be formulated about numerous, if not every, mathematical identity, this one just being more emotionally striking than most.

By Taylor series expansion, we understand why it is true. Asking for "what it means" is like asking for the meaning of life. Interesting question but not a mathematical one... Also, why just ask what this particular identity means instead of ANY and ALL other mathematical identities seems arbitrary at best, irrelevant at worst.

Regards,

Bruno

Thanks Bruno. Can you offer a creative mathematical explanation other than the usual stuff you find on-line or in textbooks? For example, the Newton's method of tangents to find the square root of a number can be explained using pure logic (without math). This was done by the Sumerians more than 5,000 years ago! There are other examples.

Funny! The base of mathematical reasoning is exactly pure logic a small set of axioms (like ZF). In that sense, you cannot really "explained using pure logic (without math)" as reasoning from first principles (i.e. with the principle of excluded middle, comparison of triangles areas, basic arithmetic operation, etc.) IS math!

There is also the idea of "proof without words" that are more like illustration of geometric equalities which can be systematize algebraically to constitute general proofs if need be. This is still mathematics by the way, just not algebra.

As in the case of the square root computation you are referring to, I think the Sumerians might have used this kind of geometrical mathematical reasoning:

https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf

I am still waiting for a creative answer to my original question (now that I have seen the responses, mathematical, graphical, etc are welcome but not those mundane proofs that can be found on the Web). I am willing to wait a few years to get an answer! As to the play on words, I have no interest in a debate that loses focus of the issue at hand. By the way, what the sumerian did was truly ingenious without the benefit of calculus but achieved the exact same result obtained using Newton’s method which requires calculus! The paper you attached didn’t cover the simplicity offered by the Sumerian. The Babylonian Semites came more than 1,500 years after the ingenious Sumerian. Thank you for the paper you attached. It was an interesting read.

The Greeks' golden era is 500 BC and the Sumerians developed their solution 3,200 BC. So your mother learned it from the Greeks who learned it from the Sumerian! This is not the point. If you can think critically without a calculator, then how would you find the square root of a number? Similarly, can you use similar methodologies used by the Sumerian to interpret Euler's identity. By the way, I use the Euler identity in dealing with buildings response to earthquakes, wind, etc. I also use it in solving problems dealing with seepage and fluid flow.

I would recommend two books, "the story of PI" and "the story of e". Interesting books on both subjects with some historical perspectives. Also, the paper you referenced doesn't do justice to what the Sumerian did in constructing tables for roots for practical use. They knew the positional notation and the Greeks and Roman didn't! My point isn't what the sumerians knew and the significance of their contributions but their informal approach is solving a mathematical problem of square roots of numbers. Is there a way to explain the Euler identity using an informal approach? Your earlier mathematical answer was quite unique and I never saw it before. The answers provided by Kåre Olaussen and Viera Čerňanová were familiar to me. Thank you for your interest in this subject.

Dear PT Followers,

Let me remind a many times stressed additional interesting feature of the Euler equation: it uses all basic operations on numbers X and Y assigning them numbers: X+Y, X.Y, XY and the most common operation X=Y assigning them a logical sentence. All these properties make the equation really very wonderful. But there is a little inconvenience: since 1 = ei 2 k \pi with arbitrary nteger k , hence one obtains NON-UNIQUENESS of the LHS: ei \pi+1 = (?) (e1+i 2 k \pi)i \pi + 1 = ei \pi - 2 k (\pi)^2 +1 \ne 0.

Obviously, this piece of bad news does not depreciate the ingenious equation! For giving it simple explanation, let me suggest the following: Let us accept a convention, that moving by ei \phi is the same as to move by a unit toward asimuth \phi. Thus, e.g. adding ei \pi /3 to e-i \pi /3 results into a move by one unit in direction "north". Obviously then, e- i \pi is a move by one unit in direction "south". In order to be back - one need to add a move by 1 in direction "north". The presented interpretation is called trigonometric, which name does not require any explanation. The main feature, however, is the multiplication rule ei \phi ei\psi= ei (\phi+\psi)which can be seen as a composition of rotations by angle \phi then \psi (same as by angle \phi + \psi), without magnification. Despite this, the strict explantion is due to mathematics, and can be obtained only as a property within a system of well defined notions. But this is not the subject of this question.

Thank you Joachim Domsta

Yes, it is already mentioned in Amir's question of this thread:

It is an identify that contains the most beautiful entities encountered in math, namely π, i, e, 0 and 1. It combines the real and the imaginary.

Thank you Joachim Domsta!

Indeed, it is easier to explain it to mathematicians. But I will try to do it for normal people.

Choose a thin tree; let it be your origin (0,0) in a complex space. Let x axis looks East and y - North. Attach yourself with a log (or rope) of 1 meter long to this tree and start from the point (1,0) in these coordinates. Then rotate 180 degrees (=π radians) counterclockwise, and find yourself in the point (-1,0), looking to the West... This is probably all that normal people can understand. But if they learned some university math, probably they will understand (at least partly) the text below.

Now think about one-to-one correspondence between your Cartesian coordinates and complex plain (apart from addition, it involves a product of 2 numbers and the equality i*i=-1) and describe your motion mathematically. This will be exactly (after you define this exponent by Taylor series of normal exponent, use i*i=-1, recall Taylor series for cos z and sin z, get e^(iz)=cos z +i sin z, and use geometrical interpretation of trigonometric functions to understand that this exponent just describes counterclockwise rotation on a unit circle):

e^(iπ) = -1.

Thank you Yuri Yegorov. I know how to prove it but not explain it. The Euler proof is well know.

From the famous Euler’s identity e^(iπ) = -1 it follows immediately the curious “equality’’ e^(-i^2π) = (-1)^(- i), i.e.,

e^π = (-1)^(-i). (1)

However, in Complex Analysis the “equality’’ (1) holds in the sense that for the value of (-1)^(-i) it is assumed the principal value which corresponds to the principal branch of the logarithm ln(-1) (with k = 0). Namely,

(-1)^(-i) = e^(-i ln (-1)) = e^(-i (πi + 2kπi)) = e^( π + 2k π), k = 0, ± 1, ± 2, ± 3,….

which for k = 0 coincides with (1).

Yet another mathematically explanation

Namely, -1 belongs to arithmetic, (complex unit) i belongs to algebra, π belongs to geometry, e belongs to analysis, and it is a wonder that all these disciplines are united by this identity.