Integer

What counts

$$\ldots, \; -3, \; -2, \; -1, \; 0, \; 1, \; 2, \; 3, \; \ldots$$

Number	Integer?
$7$	yes
$-12$	yes
$0$	yes
$1.5$	no
$\tfrac{2}{3}$	no

Every integer is a rational number, since $5$ can be written $\tfrac{5}{1}$. The reverse is not true.

What the sign records

A negative integer describes a quantity in the opposite direction from a positive one: below sea level, money owed, degrees below freezing, time before launch.

The sign is a direction, not a size. A debt of $500$ dollars is a large amount of money, and $-500$ is a long way from zero.

Opposites

Every integer has an opposite the same distance from zero on the other side, and the two add to zero:

$$6 + (-6) = 0$$

Zero is its own opposite, which is why it carries no sign — it sits on neither side of itself.

Why they were needed

Without negatives, $3 - 8$ has no answer. Extending the number line leftward makes subtraction always work, which is the same kind of move as inventing fractions so that division always works.

See what integers are for the full picture, and the sign rules for why two negatives multiply to a positive.