The Constants pi, e, i, 0, and 1.Date: 28 Jun 1995 18:40:28 -0400 From: Thomas D. Jacob Subject: e^(pi*i) A friend told me that e^(pi*i) = -1 (That is, e to the power of pi times i equals negative one.) Can you tell me anything about this? (Such as why two irrational numbers can together produce such an ordinary result.) Thanks. Thomas D. Jacob Date: 28 Jun 1995 20:35:56 -0400 From: Dr. Ken Subject: Re: e^(pi*i) Hello there! Yes, this is one of the coolest things in Math. In fact, there's an even better way to write it: i Pi e + 1 = 0 This way, you combine the five big constants in mathematics: 0,1,i,Pi, and e. You also get the three main operations: +, *, and exponentiation. And you get the notion of equality, which is so key in mathematics. To understand how this works, you have to know the fact that e^(i*t) = Cos(t) + iSin(t). This in itself is a really cool thing. What it means is that if you let t go from zero to 2 Pi, the function e^(i*t) will trace out the unit circle in the complex plane. So to rotate any complex figure by t radians, multiply it by e^(i*t). Anyway, once you have that, plug in Pi for t, and you've got your formula! Thanks for the question. -K |
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