Why does e^pi * i + 1 = 0 ?
January 6, 2008 4:23 PM Subscribe
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?
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.
posted by chndrcks at 5:42 PM on January 6, 2008
posted by chndrcks at 5:42 PM on January 6, 2008
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?
posted by LobsterMitten at 5:42 PM on January 6, 2008
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?
posted by LobsterMitten at 5:42 PM on January 6, 2008
The Heart of Mathematics? It comes highly recommended from a certain future mathematician friend of ours.
posted by martinX's bellbottoms at 5:46 PM on January 6, 2008
posted by martinX's bellbottoms at 5:46 PM on January 6, 2008
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.
posted by pammo at 5:49 PM on January 6, 2008
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.
posted by pammo at 5:49 PM on January 6, 2008
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.
posted by jewzilla at 7:11 PM on January 6, 2008
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.
posted by jewzilla at 7:11 PM on January 6, 2008
Best answer: Where Mathematics Comes From, by Lakoff and Núñez, uses Euler's identity as an example in later chapters.
posted by nebulawindphone at 8:01 AM on January 7, 2008
posted by nebulawindphone at 8:01 AM on January 7, 2008
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.
posted by Reverend John at 9:13 AM on January 7, 2008
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.
posted by Reverend John at 9:13 AM on January 7, 2008
Best answer: 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.
posted by vilcxjo_BLANKA at 1:20 PM on January 7, 2008
It's brilliant.
posted by vilcxjo_BLANKA at 1:20 PM on January 7, 2008
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posted by edd at 5:29 PM on January 6, 2008