Associative Law

In Mathematics, associative law is applied to the addition and multiplication of three numbers. According to this law, if a, b and c are three numbers, then;

a+(b+c) = (a+b)+c

a.(b.c) = (a.b).c

Thus, by the above expression, we can understand that it does not matter how we group or associate the numbers in addition and multiplication. The associative law holds only for the addition and multiplication of all the real numbers but not for subtraction and division.

Also, learn:

• Commutative Law
• Distributive Property

In Maths, associative law is applicable to only two of the four major arithmetic operations, which are addition and multiplication. But for other arithmetic operations, subtraction and division, this law is not applied, because there could be a change in result. This is due to change in position of integers during addition and multiplication, do not change the sign of the integers.

As per associative law, if we add or multiply three numbers, then their change in position or order of numbers or arrangements of numbers, does not change the result. This law is also called associative property of addition and multiplication.

Associative Law Formula

The formula for associative law or property can be determined by its definition. As per the definition, the addition or multiplication of three numbers is independent of their grouping or association. Or we can say, the grouping or combination of three numbers while adding or multiplying them does not change the result.

Let us consider A, B and C as three numbers. Then, as per this law;

A+(B+C) = (A+B)+C

A × (B × C) = (A × B) × C

Associative Law of Addition

The addition operation follows associative law, i.e. despite how numbers are combined the final sum of the numbers will be equal. If X, Y and Z are three numbers then;

X+(Y+Z) = (X+Y)+Z = X+Y+Z

Associative Law of Multiplication

The multiplication operation obeys associative law, i.e. no matter how numbers are clubbed, the final product of the numbers will be equal. If X, Y and Z are three numbers then;

X×(Y×Z) = (X×Y)×Z = X×Y×Z

Proof of Associative Law

We have learned how associative law works. Let us now prove this property with the help of examples.

Proof of Associative Law of Addition

Let us prove the associative law for addition with the help of examples.

Example 1: Prove that: 1+(2+3) = (1+2)+3

Taking LHS first,

1+(2+3) = 1+5 = 6

Now let us take RHS

(1+2)+3 = 3+3 = 6

Hence, if we compare, we get;

LHS = RHS

Therefore,

1+(2+3) = (1+2)+3.

Hence, Proved.

Example 2: Prove that: 3+(-7+9) = (3+(-7))+9

Taking LHS first;

3+(-7+9) = 3+(2) = 5

Now, taking RHS,

(3+(-7))+9 = (3-7)+9 = -4+9 = 5

Hence, on comparing, LHS and RHS are equal,

Thus, it is proved that;

3+(-7+9) = (3+(-7))+9

Proof of Associative Law of Multiplication

Now, let us prove the associative law for multiplication with the help of examples.

Example 3: Prove that:1×(2×3) = (1×2)×3

Taking LHS first,

1×(2×3) = 1×6 = 6

Now let us take RHS

(1×2)×3 = 2×3 = 6

Hence, if we compare,

LHS = RHS

Therefore,

1 × (2 × 3) = (1 × 2) × 3.

Hence, Proved.

Example 4: Prove that: 3×(-7×9) = (3×(-7))×9

Taking LHS first;

3×(-7×9) = 3×(-63) = -189

Now, taking RHS,

(3×(-7))×9 = (-21)×9 = -189

Hence, from LHS and RHS, it is proved that;

3+(-7+9) = (3+(-7))+9

Why Not Subtraction and Division?

As we have already discussed, associative law is not applicable to the subtraction and division of real numbers. Let us understand with examples.

Subtraction

Suppose, 5, 8 and 9 are three integers.

Let us assume that associative law is applicable to subtraction. Thus,

5 – (8 – 9) should be equal to (5 – 8) – 9

On solving both the expressions, we get;

5 – (8 – 9) = 5 – (-1) = 5 + 1 = 6 ………….(i)

(5 – 8) – 9 = (-3) – 9 = – 3 – 9 = -12 ………..(ii)

Clearly, from equations (i) and (ii), we get that;

6 ≠ -12

Hence,

5 – (8 – 9) ≠ (5 – 8) – 9

Therefore, our assumption is wrong and associative law is not applied to subtraction.

Division

Suppose, 27, 9 and 3 are three integers.

Let us assume that associative law is applicable to division Thus, the grouping of integers will change the result.

So,

27 ÷ (9 ÷ 3) = 27 ÷ 3 = 9 ………..(i)

(27 ÷ 9) ÷ 3 = 3 ÷ 3 = 1 ………..(ii)

Clearly, from equations (i) and (ii), we get that;

9 ≠ 1

Hence,

27 ÷ (9 ÷ 3) ≠ (27 ÷ 9) ÷ 3

Therefore, our assumption is wrong and associative law is not applied to subtraction.

Practice Questions

Group the given expression as per associative law and compare the final value.

1. 3 + 9 + 3
2. 4 + 5 + 1
3. 7 + 9 + 3
4. 3 x 9 x 2
5. 5 x 3 x 9
6. 2 x 9 x 3