Fundamental Theorems

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/