Associative Law

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 are 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

There are other two laws or properties of arithmetic operations, they are:

Associative Law Proof

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

Proof of Associative Law of Addition

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, 

LHS = RHS

Therefore,

1+(2+3) = (1+2)+3. 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, from LHS and RHS, it is proved that;

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

Proof of Associative Law of Multiplication

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. 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

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