What is an Integer?

Date: 10/03/2002 at 11:42:19
From: Patty Anderson
Subject: Integer definition

Why isn't -1/2 considered an integer? You can certainly have negative 
fraction situations in real life. My class would like this explained.

Thank you.

Date: 10/03/2002 at 12:27:13
From: Doctor Peterson
Subject: Re: Integer definition

Hi, Patty.

How has your text defined integers? They must have given a very faulty 
definition, if any, in order for the class to ask this question, but I 
suspect that many students have this confusion.

I have observed that integers are often introduced in a way that gives 
an entirely wrong impression of what they are: that any negative 
number is an integer. (They may not even understand that natural 
numbers are integers too; the set of integers extends the set of 
natural numbers by including negative numbers and zero.) To a 
mathematician, the fact that an integer can be negative is 
unimportant; what makes an integer an integer is that it is WHOLE. 
That's the root of the word "integer," which in Latin means whole (as 
in "integrity," and the related words "intact" and "entire").

Here's what's going on: in introducing the various types of numbers to 
students, schools typically start with the natural numbers, then bring 
in zero, then add fractions, and only then introduce the idea of 
negative numbers. (I believe they should be introduced long before 
that, because the concept is really very simple.) Both whole numbers 
and fractions can have a negative sign put on them, but integers are 
specifically the positive and negative whole numbers. But since, for 
some reason, negative numbers are usually introduced using the concept 
of integers (as if students had not yet learned about fractions), 
students imagine that any number with a sign ought to be an integer. 
That's not true.

A mathematician sees numbers in a different way. We may start with 
the natural numbers in an axiomatic development, but then immediately 
move to the integers, by introducing the concept of sign. (This is 
done so that any equation of the form x+a=b can be solved.) Then we 
introduce the rational numbers (fractions), so that equations of the 
form ax+b=c can be solved. Finally we introduce the irrational 
numbers (which together with the rational numbers constitute the real 
numbers), and then the complex numbers, so that any equation can be 
solved. Looking backward, once we know about all these kinds of 
numbers, the integers are simply the "whole" numbers among the reals, 
as the rationals are those real numbers which can be written as a 
fraction; the property of having a sign is nothing special. The 
natural numbers are the only set that does not have signs.

So here are the main sets of numbers your students should be aware of:

    natural numbers: 1, 2, 3, ...

    integers: ..., -3, -2, -1, 0, 1, 2, 3, ...

    rational numbers: any ratio of an integer to a natural number,
                      such as -3/5

    real numbers: the whole number line

If you have any further questions, feel free to write back.

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