The name “number theory”
can cause cognitive dissonance: it sounds as if number
theory should be the theory
of all numbers!
abstractmath.org
help with abstract math
Produced by Charles Wells. Home
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Posted 6 February 2016
INTEGERS
An integer is any whole number. An integer can be zero, greater than zero or less than
zero. The integers greater than zero are the natural numbers. I will not give a formal
definition of integer, but many of their basic properties will be spelled out here and in the
chapter on number theory.
Examples -3, 0, 42 and one million are integers.
Terminology Number theory (MW, Wik) is the branch of math concerned with the theory of the
natural numbers. Abstractmath.org has a separate chapter introducing some basic ideas about number
theory.
Properties of integers
Closure
If m and n are integers, then so are m+n, m–n abd mn. This is described by saying that the
integers are closed under addition, subtraction and multiplication. They are not closed under division.
For example, 3 and 5 are integers but 3/5 is not an integer.
Compare:
 The natural numbers are closed under addition and multiplicatrion but not subtraction.
 The rational numbers are closed under addition, subtraction, multiplication and division by a
nonzero rational.
Positive and negative
For any integer n:
 n is positive if n > 0.
 n is negative if n < 0.
 n is nonnegative if n ³ 0.
Terminology In France and possibly other countries schools may use “positive” to mean
“nonnegative”, so that zero is positive. In abstractmath.org, 0 is neither positive or negative.
Images and metaphors for integers
I will give two ways it is useful to think about integers. You will no doubt be familiar with them. They
are included here partly to illustrate the ways in which we think about mathematical objects using images
and metaphors. (In other words, in this section, I am not teaching you about the integers as much as I
am teaching you about images and metaphors!)
Ordered list infinite in both directions
The positive integers increase in size to the right and the negative integers increase in size to the
left:
… –5
–4
–3
–2
–1
0
1
2
3
4
5
…
In contrast to the natural numbers they form a list that is infinite in both directions.
Adding a positive number n to another one takes you to the number n spaces to the right. Adding a
negative number –n takes you to the number n spaces to the left. This is a special case of the way
positive and negative distances are represented on the real line. (Warning: See minus.)
Credit and debit
A negative integer such as –5 does not have an immediately obvious interpretation as the number of
elements of a set. One interpretation that does make sense of negative integers involves credit and
debit. If you have n dollars (for n positive) that is what you have. Having –n dollars represents the fact
that you owe n dollars.
Credit and debit for substances requires the real numbers.