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Is There a Universal Set of All Numbers?

Date: 06/16/2004 at 13:56:47
From: John
Subject: The universal set of all numbers.

Dear Dr. Math,

The real numbers (including all its subsets) and the imaginary numbers 
are subsets of the complex numbers.  Is the set of complex numbers a 
subset of a more universal set?  Is there a universal set of all 
numbers agreed upon today?

New numbers are created to help answer new problems.  Has there been a 
need to create a set of numbers other than the complex numbers?  I 
have not been able to find an answer to this question from texts.



Date: 06/16/2004 at 21:26:31
From: Doctor Vogler
Subject: Re: The universal set of all numbers.

Hi John,

Thanks for writing to Dr. Math.  Interesting question.  There are
different ways to create "new numbers," and I'll describe them briefly
for you.  I'll start with the notion of a mathematical "field"--this
is a word that describes any collection of numbers that satisfies
several basic rules, including closure under addition, subtraction,
and multiplication (which means that you can add, subtract, or
multiply any two numbers in the collection and get another number in
the collection), closure under division by nonzero numbers, the
commutative laws of addition and multiplication, the distributive law,
the additive identity 0 (zero), and the multiplicative identity 1 (one).

For example, the collection of integers is not a field; can you say
what rule it breaks?  The collection of rational numbers, on the other
hand, IS a field.  The collection of real numbers is also a field. 
And so is the collection of complex numbers.  But there are others.

First of all, there are "field extensions."  We can do the four 
functions in a field and always stay in the field, but sometimes
polynomials don't have all their roots in the field.  For example, the
polynomial

  x^2 - 2

has no root in the field of rational numbers.  We can make an 
extension of the rational numbers by also including the square root of
2.  In fact, when we do that, we get all numbers of the form

  r + s*sqrt(2)

where r and s are rational, and this makes a field.  (Can you write
1/(1 + sqrt(2)) in the above form?)

The complex numbers are what you get when you include a root of

  x^2 + 1

in the field of real numbers.  All such field extensions of the 
rational numbers or of the real numbers are contained in the field of
complex numbers, and every polynomial has all of its roots in the
complex numbers.  This kind of a field is called "algebraically
closed," and in this sense there is no "bigger" field than the complex
numbers.

However, there are very different kinds of fields.  For example, you
can make so-called transcendental (or infinite) extensions of fields.
Instead of adding a root of a polynomial, you can add a "number" (or
symbol, like x) that is the root of no polynomial, and then you end up
with the field of all polynomials and rational functions (polynomial
divided by a polynomial) in that symbol (x).  The field of rational
functions in x with complex coefficients is, therefore, a field
"bigger" than the complex numbers (in the sense that it contains the
complex numbers), but these aren't "numbers" like you normally think
of numbers.

A little more like "normal" numbers are the finite fields.  Finite
fields have a "characteristic" which is a prime number, and this prime
number is like another zero in this field.  For example, in the finite
field of 7 elements, which has characteristic 7, there are only 7 
numbers, namely 0, 1, 2, 3, 4, 5, and 6.  You can add, subtract,
multiply and divide by nonzero in this field, and the rule is that 7 
is the same as 0.  (This is arithmetic modulo 7, if you know what that
means.)  For example, 1 + 6 = 0, which means that -1 = 6.  And 2*4 =
1, which means that 1/2 = 4.  Then you can make extensions of these by
including a root of a polynomial.  And you can make transcendental
extensions, too.  These are sometimes useful on computers, since
computers can only represent finitely many numbers in a fixed amount
of memory, it can be useful in some applications to use the finite
field of 2^n elements for some number n.

Finally, there is a very different kind of numbers called the p-adics.
They also make a field, but these get really complicated, and they
are not really related to the other fields at first sight (until you
learn about "local fields" but that's getting pretty complicated).  If
you're interested, you can read a thread about the p-adics from our
archives:

  An Introduction To P-adic Numbers
    http://mathforum.org/library/drmath/view/65286.html 

If you have any questions about this or need more help, please write
back, and I will try to offer further thoughts.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Imaginary/Complex Numbers
College Logic

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