Euler's Equation: First Step
Date: 05/18/99 at 09:46:18 From: Max Shaffer Subject: Euler's equation (e^pi*i=-1) I've read your section on Euler's equation. http://mathforum.org/dr.math/faq/faq.euler.equation.html I have a question on the very first step: any complex number, z, equals cos(x)+isin(x). Why is this? Is this a definitional thing, that any complex number can be expressed in this fashion?
Date: 05/18/99 at 09:57:30 From: Doctor Mitteldorf Subject: Re: Euler's equation (e^pi*i=-1) Dear Max, Of course it's not true that any complex number can be expressed as cos(t)+i*sin(t). For example, the number 2 is a complex number that cannot, because there's no x for which cos(t) = 2. Any complex number with UNIT MODULUS can be expressed as cos(t)+i*sin(t), i.e., any complex number that obeys |z|^2 = 1. So starting with any z at all, you can first write it as R times z', where z' has unit modulus, and R is the modulus |z|. Then it is true that z' can be expressed as cos(t)+i*sin(t). Just let x be the angle whose tangent is the quotient of the imaginary part of z to the real part of z. In other words, take the imaginary part of z and the real part, and divide them. Think in terms of the complex plane where the y axis represents the imaginary part and the x axis the real part. The angle t is formed when you connect the point x+iy to the origin. The length of the connection is R=|z|. Basic trig definitions give the imaginary part as R sin(t) and the real part as R cos(t). - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
Date: 05/18/99 at 10:11:05 From: Doctor Rob Subject: Re: Euler's equation (e^pi*i=-1) Thanks for writing to Ask Dr. Math! I suppose you are talking about the second (or calculus) proof of Euler's Equation. No, it is not true that any complex number can be written in the form cos(x) + i*sin(x). That is not what is stated there. What is meant is this: let z be a shorthand name for the expression cos(x) + i*sin(x). If you look at the section entitled "An Application," you will see how to express any complex number z as r*[cos(t) + i*sin(t)], where t and r = |z| = sqrt(a^2+b^2) are real numbers. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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