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### Fundamental Theorems

```
Date: 10/02/2000 at 17:53:56
From: Chris Rez
Subject: Fundamental theorems

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/
```
Associated Topics:
High School Number Theory

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