Euler’s Identity, Part One

Euler’s identity is the equation:


\[
e^{i \pi} +1=0.
\]

If you have any taste for mathematics at all, it is hard not to smile at this. In one equation we have each of five “special” numbers (e, i, pi, 1 and 0), along with one instance of each of three basic arithmetic operations (addition, multiplication and exponentiation.) Not too shabby!

But why is the equation true? Well, the first thing to notice is that we have an imaginary number in the exponent. That means that appreciating this equation requires an understanding of how we deal with such things. Towards that end, we have the following definition:


\[
e^{i \theta}=\cos \theta+ i \sin \theta,
\]

where theta is some real number.


Given this definition, it follows that


\[
e^{i \pi}=\cos \pi+ i \sin \pi.
\]

If you remember your basic trigonometry, then you know that


\[
\cos \pi = -1 \phantom{xxx} \textrm{and} \phantom{xxx} \sin \pi=0.
\]

Euler’s equation now follows immediately.

But now you might be thinking that the fix is in. Anyone can whip up a bizarre definition and then discover that, as a result, one arbitrary collection of symbols is equal to another. Why, though, would you summon forth such a bizarre definition in the first place? Who said you could drag trig into this?

Appreciating the beauty of Euler’s identity requires understanding why that triggy definition is the appropriate one to use. And understanding that requires thinking for a moment about what we want out of a definition.

So let’s go back to the natural numbers, by which I mean


\[
1, 2, 3, 4, 5, 6, 7, \dots
\]

Useful stuff, no question about it. You can use these numbers to label collections of objects. You can even do arithmetic with them, so long as you are content with addition and multiplication. Subtraction, alas, is more difficult. You are fine if the number you are subtracting is smaller than the number being subtracted from, (if the subtrahend is smaller than the minuend, in the jargon) but you are SOL if the smaller number is on the left.

It would be nice if we could embed the natural numbers within a larger set. If we play our cards right we would then have a set that preserves all of the good stuff about natural numbers, but wold also give us some added functionality. There is such a set, of course, and it is called the integers. It is the set you obtain when you take the natural numbers, and throw in these symbols:


\[
0, -1, -2, -3, -4, -5, -6, \dots
\]

The trouble is that you can’t just throw in symbols willy-nilly. You have to verify that your new symbols play well with others. Can I do arithmetic with my new numbers in a way that is consistent with everything else I know about arithmetic? The answer is yes, but only if I define things carefully. For example, the somewhat awkward seeming definition that the product of two negative numbers is positive turns out to be logically required. I’ll save the details for another post.

The process continues. The integers are nice, but in general you can’t divide with them. So we embed them inside the rational numbers (the term “rational” indicating that we are talking about fractions, or ratios), and now we can divide. In the rational numbers we can add, subtract, multiply and divide, but they still lack certain desirable properties related to the convergence of sequences. So we embed the rationals inside the real numbers.. The real numbers are pretty good, you can even do calculus with them, but tit turns out that certain polynomials do not have solutions. So the real numbers get embedded within the complex numbers.

Which brings us back to complex exponentiation. We will take for granted that we know how to raise a real number like e to a real number exponent. We shall then insist that our definition of complex exponentiation be consistent with that prior definition for real exponentiation. At a minimum, if we take x+iy to denote an arbitrary complex number, then we better have:


\[
e^{x+iy}=e^x e^{iy},
\]

since this is a consequence of one of our standard rules for manipulating exponents. And since we already know what the first term in the product means, our problem has been reduced to considering what we might call “pure imaginary numbers.”

Now what? For real exponentials we started with positive integer exponents. Raising a number to such a power meant multiplying the number by itself the prescribed number of times, a simple and common sensical notion. It turns out that the definitions for negative, rational and real number exponents are then forced on us logically, just like we were forced to define the product of two negative numbers to be a positive number. The details of how that is done require a post of its own.

But now we seem to have hit a wall. How on earth are we going to extend this definition to imaginary exponents? There’s a story in that, but we shall save it for the dramatic conclusion…

Comments

  1. #1 Traces
    December 27, 2011

    cos pi =-1
    first!!!

  2. #2 Jason Rosenhouse
    December 27, 2011

    D’oh! I’ve corrected the error. Thanks for pointing it out.

  3. #3 DerekM
    December 27, 2011

    I was first introduced to this equation as a teenager in a book of Martin Gardner’s Scientific American columns. I was fascinated by it then as i was only just starting to get my head around trigonometry. Thanks for reintroducing it to me.

  4. #4 Lxndr
    December 27, 2011

    Using pi and not tau? tsk tsk.

  5. #5 Peter
    December 28, 2011

    Lakoff and Nunez wrote a book, Where Mathematics Comes From, about how we think about math in terms of cognitive science. It started from our intuitive notions of numbers and arithmetic, how that gets generalized into more formal and abstract concepts, and ended up with four case studies building up to all the concepts we use when thinking about Euler’s Identity.

  6. #6 Peter
    December 28, 2011

    Also, integer division is very useful. What we really get from the rationals is multiplicative inverses.

  7. #7 Lou Jost
    December 28, 2011

    Peter (#6), integer division is not closed on the set of integers (1/2 is not an integer), so what we really get from the rationals is closure.

  8. #8 Conrad Halling
    December 28, 2011

    I recently finished reading “An Imaginary Tale: The Story of [the Square Root of Minus One]“, written by Paul J. Nahin (http://press.princeton.edu/titles/6388.html), which provides a thorough and entertaining background on the history of i and how it is used in a variety of fields, including electrical engineering. One of the surprises from the book is that i to the ith power is a real number, not an imaginary number.

    The book is written for lovers of math, with formulas or equations on nearly every page. My only complaint is that the figures, which were created manually by a scientific illustrator, are not always accurately drawn.

  9. #9 Peter
    December 28, 2011

    Integer division is closed on the set of integers. a / b = c where c is the greatest integer s. t. b * c &lt= a. And you can get an integer remainder R if you want, where R = a – b * c. Everyone knows this operation, and it’s widely used even past elementary school. Math students study it in abstract algebra classes.

  10. #10 Tau
    December 28, 2011

    How is this any more “beautiful” than 0 + 1 + 2 = 3? It’s just a coincidence based on an incorrect value of pi = 3.14 and adding 1 to both sides. It should be based on 6.28, making the identity e^iτ = 1, rather than e^iπ = −1.

  11. #11 Physicalist
    December 28, 2011

    I always was a bit suspicious of that whole smuggling-in-waves-when-all-we’ve-got-is-good-old-fashioned-multiplication thing.

  12. #12 Jason Rosenhouse
    December 28, 2011

    DerekM –

    Martin Gardner introduced a lot of people to interesting mathematics. Glad you liked the post, and stay tuned for part two.

    Peter –

    I was not aware of the book by Lakoff and Nunez. Thanks for calling it to my attention. It sounds like an interesting read.

    Also, while it’s true that in some cases you can divide one natural number by another and obtain a natural number as your answer (for example: 10 divided by 2 = 5), my point, as Lou Jost points out, is that you can’t always do that. For example: 10 divided by 3 is not a natural number. You can certainly obtain a quotient and a remainder, and that is useful in many circumstances, but it is not the same as embedding the natural numbers within a large set in which the division of one number by another produces something else in the set. (Which is to say that the set is closed under division.)

    Conrad Halling –

    I’ve been meaning to read Nahin’s book for some time, but haven’t yet gotten around to it. I’ve heard that Barry Mazur’s book, Imagining Numbers, is also quite good.

    Lxndr –

    I’m afraid I’ve never seen anyone use tau instead of pi in this context.

    Tau –

    I think the beauty comes from the surprise value of the equation. The numbers e, i, pi, 1 and 0 all have precise definitions that certainly do not appear to have anything to do with one another. But it turns out they are all related by a simple equation. I find that cool.

  13. #13 Peter
    December 28, 2011

    I’m not making this up! And here’s another definition that doesn’t refer to regular division at all (sec 1.2.12), that’s basically the same as the definition I gave above. This well known operation maps Z x Z -> Z for all elements of Z x Z.

  14. #14 AnswersInGenitals
    December 28, 2011

    What Euler’s formula is telling us is that if you take one step forward (the +1) and then turn 180 degrees (the e(ixpi)) and then take one step back (the 1 in 1xe(ixpi), which in common notation is considered to be understood), you get back to exactly where you started from (the =0). Not too startling, but the most obvious and simple discoveries always prove to be the most useful and powerful. This supports the contention that math is a fundamentally empirical endeavor.

  15. #15 phayes
    December 28, 2011

    @Peter and Lou. Since zero is an element, even the set of rationals isn’t closed under (redefined) division.

  16. #16 joemac53
    December 28, 2011

    @Conrad, you mentioned that the manual drawings in a book were not always accurate. I run into this all the time. What do people use to draw circles, axes, vectors, etc. with units and coordinates? Real cut and paste is a time sink.
    Is there a true scientific word processor with drawing?
    (I used Mathwriter for Mac for 18 years, but can’t get around its age any longer)

    Love this post.

  17. #17 Peter
    December 28, 2011

    @15: well, true. Although that’s not actually a problem for the mu-recursive definition in that second link. a / 0 = a + 1 for that definition. But of course, that’s not usually what we want even when when all we need is integer division.

    And is integer division a “redefinition”? Surely people used integer division before the rationals were invented. I mean, the (actually closed on Z!) mu-recursive definition is new. But the division algorithm is old.

  18. #18 Jason Rosenhouse
    December 28, 2011

    joemac –

    Love this post.

    Thanks!

  19. #19 Another Matt
    December 29, 2011

    A pro-tau argument: http://tauday.com/#sec:euler_s_identity

    I agree with Peter about integer division – his point is that there’s a perfectly good arithmetic operation to divide all integers with an integer result (e.g. with ints on a computer), but for division to be useful in an algebra (rather than just as an arithmetic tool) you need the inverse operation, hence the rationals.

    Incidentally this reminds me that defining “division” as “inverse of multiplication” has some interesting effects in modular integer arithmetic as well. It has some practical application in western music theory, and there’s a kind of controversy in the music theory world about the sense in which music theory and analysis (and cognition) is to be regarded as an empirical science (to tie this to the scientism posts).

  20. #20 Jason Rosenhouse
    December 29, 2011

    Peter and Another Matt –

    I agree with Peter about integer division – his point is that there’s a perfectly good arithmetic operation to divide all integers with an integer result (e.g. with ints on a computer), but for division to be useful in an algebra (rather than just as an arithmetic tool) you need the inverse operation, hence the rationals.

    OK, I see your point now. But in my defense I think it’s pretty common usage to say simply, “You can’t divide in the natural numbers,” rather than, “Except for 1, natural numbers don’t have multiplicative inverses,” or some such.

  21. #21 Pseudonym
    December 31, 2011

    I’m also in the class of people who have a deep appreciation for all things mathematical but really don’t find Euler’s identity in any way remarkable or interesting.

    The complex exponential formula, expanding e^(iθ) in terms of cos θ and sin θ, now that’s interesting. Euler’s identity is almost tedious by comparison.