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What is the Definition of 'Number'?

Date: 01/26/2006 at 11:24:55
From: Steven
Subject: Definition of "number"

Dr. Math,

I'm at an algebra/trigonometry level of mathematics.  I feel silly 
asking this, but can you tell me the definition of "number"?

I read the following response to a commentary on imaginary numbers  
and realized that I too have the same concept of what a number is, 
but I'm not sure it's correct and that may be the "block" that I run 
into occasionally.  Here it is:

"One of the problems with the concept of "i" as a number is that most
people associate the word "number" with the concept of a measure of
the magnitude of some set, such as the number of people in a stadium.
Since one cannot say there are "2+i" slices of bread in a loaf, people
have a bad reaction to calling "i" some sort of number."

This statement implies that while numbers are often used as a concept 
of measurement, that is not their true purpose.  Did I misunderstand?
What DOES a number really represent?



Date: 01/26/2006 at 12:50:07
From: Doctor Peterson
Subject: Re: Definition of 

Hi, Steven.

I think the key here is to realize that we often start with a basic, 
common-sense concept, and then extend the definition to cover 
something broader.  The basic idea of a number that children develop 
starts with mere counting, but soon extends to the more general idea 
of measuring (2 1/2 feet, for example), which is probably where most 
people stop, because that's enough for most everyday uses of number.
 
Those who continue in math continue to extend the concept further; 
ultimately a number is just anything we choose to call a number, 
because it behaves like a number.  Complex numbers are the ultimate 
extension in the development that starts with natural numbers, then 
includes negative numbers, fractions, irrational numbers, and 
finally imaginary numbers.  Each extension does not change the basic 
behavior of numbers (we can still add, multiply, and so on), but lets
us do more--for example, once we have fractions, we can divide any
two numbers (except for division by zero); once we have complex
numbers, we can take the square root of any number.  There are other
kinds of "numbers" that are more abstract than this, and not strictly
an extension of ordinary numbers in that sense; but these are still
called numbers because you can do the same operations on them.

You may notice that I've avoided actually giving a single definition 
of "number"!  That's because we really don't need to do that; we just 
take the basic term and apply it to more and more abstract objects on
the basis of analogy to the numbers we already know.  Just for fun, I
looked up "number" in the Merriam-Webster dictionary (m-w.com) and
found these definitions as part of a long entry:

  1 c (1) : a unit belonging to an abstract mathematical system and
            subject to specified laws of succession, addition, and
            multiplication; especially : NATURAL NUMBER

      (2) : an element (as ) of any of many mathematical systems
            obtained by extension of or analogy with the natural
            number system 

I think that says what I'm saying, pretty concisely.

Does that help?


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
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