Comments on: Why does e^pi * i + 1 = 0 ?

Question: Why does e^pi * i + 1 = 0 ?

phrontist — Sun, 06 Jan 2008 16:23:59 -0800

I heard about a book that covered philosophical questions at the core of mathematics. This interests me, but I'm really interested because if I remember correctly it used understanding of euler's identity as an example. What was this book?

By: edd

edd — Sun, 06 Jan 2008 17:29:59 -0800

I don't know that 'philosophical questions' applies, but What Is Mathematics by Courant and Robbins (and later, but in a very different style I feel, Stewart) is a bit of a classic, which I think would include that.

By: chndrcks

chndrcks — Sun, 06 Jan 2008 17:42:42 -0800

This probably isn't the exact one you're looking for (although, I haven't finished it yet, so I'm not sure), but you might be interested in Thinking about Mathematics by Shapiro. It"s a good secondary source for the philosophy of mathematics and covers a lot of the key concepts.

By: LobsterMitten

LobsterMitten — Sun, 06 Jan 2008 17:42:53 -0800

There are a lot of philosophy of math books, and a small cottage industry of lighter math "musings" books in the last few years... I'll run through my bookshelf later to see if I can find one that fits. But:
Do you remember anything else about it? Was it for a popular-nonfiction audience, or was it an academic book? Did it just come out in the last few years? Remember where you heard about it? anything else?

By: martinX's bellbottoms

martinX's bellbottoms — Sun, 06 Jan 2008 17:46:54 -0800

The Heart of Mathematics? It comes highly recommended from a certain future mathematician friend of ours.

By: pammo

pammo — Sun, 06 Jan 2008 17:49:30 -0800

mr pammo here:

this may be what you are looking for. other books referenced on this page may be of interest as well.

that it equals 1 is just a mathematical identity. Far from any great mystical revelation, it just illustrates how concise and consistent mathematics can be. e and pi are transcendental numbers and i is imaginary, yet euler discovered years ago a simple mathematical relationship between them holds.

as to why, there are no whys, only perspectives. The result follows from a simple taylor expansion of e, or from an understanding of how exponentiation in the complex plane is equivalent to a rotation.

By: jewzilla

jewzilla — Sun, 06 Jan 2008 19:11:49 -0800

One representation for e^x is a an infinite series:

exp(x) = 1 + x + x^2/2! + x^3/3! + ...

One representation for pi is an infinite series:

pi = 4 * (1 - 1/3 + 1/5 - 1/7 + ...)

It happens that exp(pi * i), if you work it out, leaves -1. The exact method for doing this is left as an exercise to the reader, but I found this very satisfying when I first saw it.

By: nebulawindphone

nebulawindphone — Mon, 07 Jan 2008 08:01:41 -0800

Where Mathematics Comes From, by Lakoff and Núñez, uses Euler's identity as an example in later chapters.

By: Reverend John

Reverend John — Mon, 07 Jan 2008 09:13:03 -0800

The result follows from a simple taylor expansion of e, or from an understanding of how exponentiation in the complex plane is equivalent to a rotation.

Could you expand on that or point to a reference that does? The meaning of a complex exponent is something that has always bugged me.

I get the taylor expansion explanation for euler's formula, and hence the identity, but I've never been able to form an intuitive concept of what a complex exponent means.

By: vilcxjo_BLANKA

vilcxjo_BLANKA — Mon, 07 Jan 2008 13:20:54 -0800

You want the book Proofs and Refutations by Imre Lakatos. ISBN: 0521290384. The interesting thing is not that it presents a proof, but that it presents a series of proofs, one by Cauchy, that stood for long periods of time until someone noticed that the definitions maybe were wrong, or that someone had made an assumption that was maybe not right.

It's brilliant.