Laws of arithmetic

In algebra we are doing arithmetic with just one new feature − we use letters to represent numbers. Because the letters are simply stand-ins for numbers, arithmetic is carried out exactly as it is with numbers. In particular, as already explored, the laws of arithmetic (commutative, associative and distributive) hold when a and b are any numbers at all.

Commutative laws

Addition and multiplication are both commutative. This means that

3 + 4 = 4 + 3 and 3 × 4 = 4 × 3

In general,

a + b = b + a and a × b = b × a for every pair of numbers a and b

Commutativity means that we do not have to worry about whether we calculate

a + b or b + a because the answer is the same.

Similarly, calculating a × b and b × a gives the same result.

Associative laws

Addition and multiplication are both associative. This means that

6 + (4 + 2) = (6 + 4) + 2 and 6 × (4 × 2) = (6 × 4) × 2

In general,

a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c for every three numbers a, b and c

Associativity ensures that the expressions a + b + c and a × b × c are unambiguous, because it makes no difference which of the two operations is calculated first.

Example 1

Billy had 16 oranges, was given four more by Sally and then eight more by Rosie.

We can write this as 16 + (4 + 8) = 16 + 12 = 28

or (16 + 4) + 8 = 20 + 8 = 28.

Distributive laws

Commutativity and associativity are properties of a single operation. The equation 3 × (2 + 4) = (3 × 2) + (3 × 4) is an example of the distributivity of multiplication over addition. In general,

a × (b + c) = (a × b) + (a × c) for any numbers a, b and c

We can distribute multiplication over addition from the right, so

(a + b) × c = (a × c) + (b × c) for any numbers a, b and c

We can distribute multiplication over subtraction from both the left and the right, so

a × (b − c) = (a × b) − (a × c), and

(a − b) × c = (a × c) − (b × c) for any numbers a, b and c

All the above are called distributive laws.

The distributivity of multiplication over addition and subtraction is the key to the multiplication and division algorithms.

Note that we can distribute division over addition from the right, in the sense that

(80 + 20) ÷ 8 = 80 ÷ 8 + 20 ÷ 8

In summary,

Commutative law	a + b = b + a	a × b = b × a
Associative law	(a + b) + c = a + (b + c)	(a × b) × c = a × (b × c)
Distributive law	a × (b + c) = a × b + a × c	(a + b) × c = a × c + b × c