The natural numbers are enough for counting, but subtraction can produce values outside that set. While $5-3=2$ is natural, $3-5$ is not. To represent such differences, the natural numbers are extended by adding negative numbers. The resulting set is the set of integers, written $\mathbb Z$:

$$\mathbb Z=\lbrace \ldots,-3,-2,-1,0,1,2,3,\ldots\rbrace.$$

The integer number line

On the number line, positive integers lie to the right of zero and negative integers to the left.

←────|────|────|────|────|────|────|────→
    -3   -2   -1    0    1    2    3

Opposites and subtraction

Numbers the same distance from zero but on opposite sides of it are opposites. For example, $3$ and $-3$ are opposites.

Every integer $a$ has an opposite, written $-a$. Adding a number to its opposite gives zero:

$$a+(-a)=0.$$

This makes subtraction part of ordinary addition. Subtracting $b$ is the same as adding its opposite:

$$a-b=a+(-b).$$

For example,

$$3-5=3+(-5)=-2.$$

Addition, subtraction and multiplication of integers always give another integer. The integers therefore provide a number system in which positive quantities, zero and negative quantities can all be represented and compared.