Date: 10/02/2000 at 17:53:56 From: Chris Rez Subject: Fundamental theorems Sir or Madam, I really need a brief explanation of the fundamental theorems of algebra and arithmetic. I completely understand the fundamental theorem of calculus, but seem to be lost on the others. Thanks for your help and time, Chris Rez
Date: 10/02/2000 at 18:06:55 From: Doctor Schwa Subject: Re: Fundamental theorems The fundamental theorem of arithmetic says that every positive integer can be factored in one and ONLY one way into primes, such as 30 = 2*3*5. If you think this is really obvious, that's perfectly normal. Here's an example where it's not so obvious, though: the EVEN integers. In fact, it's not only not obvious, it's not true. For example, 60 can be factored into "primes" two ways: 2*30 and 6*10. 6 and 10 are prime in the set of even integers because they can't be factored into other even integers. The fundamental theorem of algebra is similar; it says that every polynomial (with real or complex coefficients) can be factored completely into complex number factors. Or, in other words, for any polynomial p(x) with real or complex coefficients, there is some complex number z so that p(z) = 0. Here's why that is important: When you start out with the positive integers, you need to invent negative numbers to solve things like x + 3 = 2. You need to invent rational numbers for things like 3x = 2. You need to invent irrationals for things like x^2 = 2, and complex numbers for things like x^2 = -1. The amazing thing is, after that you don't need to invent any more numbers to solve equations like x^2 = i; any polynomial equation with complex numbers can be solved using other complex numbers. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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