integer
Imagine a thermometer or an elevator panel: floors go up above the ground floor (1, 2, 3) and down below it (−1, −2, −3), with the ground floor itself as 0. The integers are exactly these numbers — the whole amounts in both directions, positive and negative, plus zero in the middle.
Formally, an integer is a whole number or its negative, including zero: …, −3, −2, −1, 0, 1, 2, 3, … . There are no fractions and no decimal parts. Negatives let us record debts, temperatures below zero, and positions to the left of a starting point — anything with a sense of opposite direction.
The integers fix a flaw of the whole numbers: they are closed under subtraction, so any difference of integers is again an integer (this is exactly why negatives were invented). They are still not closed under division — 7 ÷ 2 is not an integer — which is the gap that the rational numbers later fill. Every integer has an additive inverse inside the set, namely its negative.
If the temperature is 5 degrees and drops by 8, the result is 5 − 8 = −3 degrees — an integer below zero.
Subtraction can carry you below zero into the negative integers.
The set of integers is written Z (from the German Zahlen, “numbers”). Every whole number is an integer, and every integer is a rational number.