The commutative, associative and distributive laws

The commutative, associative and distributive laws

We all know that 6 X 5 = 30 and that 5 X 6 = 30, so is it true that when multiplying any two numbers together, the result is the same no matter what the order? The answer is yes! The above relationship may be stated as:

The product of two real numbers is the same no matter in what order they are multiplied. That is, ab = ba this is known as the commutative law of multiplication.

If three or more real numbers are multiplied together, the order in which they are multiplied still makes no difference to the product, for example 3 X 4 X 5 = 60 and 5 X 3 X 4 = 60. This relationship may be stated formally as:

The product of three or more numbers is the same no matter in what manner they are grouped. That is, a(bc) = (ab)c this is known as the associative law of multiplication.

These laws may seem ridiculously simple, yet they form the basis of many algebraic techniques, you will be using latter!

We also have commutative and associative laws for addition of numbers, which by now will be quite obvious to you, here they are:

The sum of two numbers is the same no matter in what order they are added. That is, a + b = b + a. This is known as the commutative law of addition.

The sum of three or more numbers is the same no matter in what manner they are grouped. That is, (a + b) + c = a + (b + c). This is known as the associative law of addition.

You may be wondering where the laws are for subtraction. Well you have already covered these when we considered the laws of signs. In other words, the above laws are valid no matter whether or not the number is positive or negative. So for example, –8 + (16 – 5) = 3 and (– 8 + 16) – 5 =3.

In order to complete our laws we need to consider the following problem. 4(5 + 6) = ? We may solve this problem in one of the following two ways, firstly by adding the numbers inside the brackets and then multiplying the result by 4, this gives: 4(l l) = 44. Alternatively we may multiply out the bracket as follows: (4 X 5) + (4 X 6) = 20 + 24 = 44. Thus whichever method we choose, the arithmetic result is the same. This result is true in all cases, no matter how many numbers are contained within the brackets!

So in general, using literal numbers we have:

a(b + c) = ab + ac

This is the distributive law.

In words it is rather complicated:

The product of a number by the sum of two or more numbers is equal to the sum of the products of the first number by each of the numbers of the sum.

Now, perhaps you can see the power of algebra in representing this law, it is a lot easier to remember than the wordy explanation!

Remember that the distributive law is valid no matter how many numbers are contained in the brackets, and no matter whether the sign connecting them is a plus or minus. As you will see later, this law is one of the most useful and convenient rules for manipulating formulae and solving algebraic equations.

Example 4.3

If a = 4, b = 3, and c=7, does a(b – c) = ab – ac

The above expression is just the distributive law, with the sign of one number within the bracket, changed. This of course is valid since the sign connecting the numbers within the bracket may be a plus or minus. Nevertheless we will substitute the arithmetic values in order to check the validity of the expression.

Then:                                      4(3 – 7) = 4(3) – 4(7)

                                                  4(– 4)   = 12 – 28

                                                   – 16    = –16

So our law works irrespective of whether the sign joining the numbers is positive or negative.

Test your knowledge 4.1

1. 6, 7, 9, 15 are ____ numbers.

2. , ,  are ____numbers.

3. Rewrite the numbers 5, 13, 16 in the form a/b, where b = 6.

4. Express the negative integers –4, –7 and –12 in the form alb, where b = 4.

5.  Can be expressed as a positive ____. It is ____.

6.  Cannot be expressed as a ____  number; it is, however, a ____.

7. Express as non–terminating decimals: (a)  , (b)  , and (c) 2.

8. Find the value of

(a) a(b+c– d), where a = 3, b = 4, c = 6 and d = –1

(b) (21 – 6 + 7)3

(c) 6x4+5x3.

9. Which of the following has the largest absolute value, –7, 3, 15, –25, –31?

10. –16+(–4)–(–3)+28= ____?

11. Find the absolute value of – 4 x (14 – 38) + (–82).

12. What is (a) , (b)

13. What is (a) (–3)(–2)(16) and (b) –3 x –2(15)?

14. Evaluate 2a(b + 2c + 3d), when a = 4, b = 8, c = –2, and d=2.