What are integers?
Integers are the whole numbers together with their negatives:
…, −3, −2, −1, 0, 1, 2, 3, …
Negative numbers describe quantities below a reference point: temperatures below 0 °C, depths below sea level, a loss in business, or money owed. Zero is neither positive nor negative.
On a number line, numbers increase as you move right and decrease as you move left. So −2 > −5, because −2 lies to the right of −5.
Addition on the number line
To add a positive integer, move right. To add a negative integer, move left.
3 + 4: start at 3, move 4 right, land on 7.
3 + (−5): start at 3, move 5 left, land on −2.
(−2) + (−4): start at −2, move 4 left, land on −6.
Shortcut rules:
Same signs: add the numbers and keep the sign. (−7) + (−8) = −15.
Different signs: subtract the smaller number from the larger (ignoring signs) and take the sign of the larger. (−12) + 5 = −7.
Subtraction: add the opposite
Every integer has an additive inverse (its opposite) that adds with it to give 0. The additive inverse of 6 is −6, and of −9 is 9.
To subtract an integer, add its additive inverse:
a − b = a + (−b)
5 − 8 = 5 + (−8) = −3
4 − (−6) = 4 + 6 = 10
(−3) − (−7) = −3 + 7 = 4
Subtracting a negative number gives a larger result. Think of it as a debt being removed.
Multiplication and division: sign rules
Signs | Result | Example |
|---|---|---|
positive × positive | positive | 4 × 3 = 12 |
positive × negative | negative | 4 × (−3) = −12 |
negative × positive | negative | (−4) × 3 = −12 |
negative × negative | positive | (−4) × (−3) = 12 |
The same rules apply to division: (−20) ÷ 4 = −5 and (−20) ÷ (−4) = 5.
For a product of many integers, count the negative factors:
an even number of negative factors gives a positive product,
an odd number gives a negative product.
So (−1) × (−1) × (−1) = −1, while (−1) × (−1) × (−1) × (−1) = 1.
Properties of integers
Property | Addition | Multiplication |
|---|---|---|
Closure | the sum of two integers is an integer | the product of two integers is an integer |
Commutative | a + b = b + a | a × b = b × a |
Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
Identity | a + 0 = a | a × 1 = a |
Subtraction and division are not commutative: 5 − 3 ≠ 3 − 5. Division of integers does not always give an integer (7 ÷ 2 = 3.5), so integers are not closed under division. Division by zero is not defined.
The distributive property links multiplication and addition: a × (b + c) = a × b + a × c. It makes mental maths easy:
16 × 102 = 16 × (100 + 2) = 1600 + 32 = 1632
A real-life example
A diver is at 15 m below sea level (−15 m). She rises 6 m and then dives 10 m deeper. Where is she now?
−15 + 6 + (−10) = −19, which is 19 m below sea level.