#Arithmetic Laws

The most common arithmetic laws are the commutative, associative, distributive, identity, and inverse laws.

• The commutative law says that changing the order of numbers does not change the answer in addition and multiplication.
• 3 + 5 = 5 + 3
• 2 × 4 = 4 × 2

• The associative law says that when you are adding or multiplying, you can change the grouping of the numbers without changing the result.
• (2 + 3) + 4 = 2 + (3 + 4)
• (2 × 3) × 4 = 2 × (3 × 4)

• The distributive law connects multiplication with addition or subtraction. It tells you that multiplying a number outside the brackets by each number inside gives the same result as multiplying first and then adding or subtracting.
• 3 × (4 + 2) = (3 × 4) + (3 × 2)

• The identity law describes numbers that leave a value unchanged.
• Adding 0 keeps a number the same: 7 + 0 = 7
• Multiplying by 1 keeps a number the same: 7 × 1 = 7

• The inverse law describes pairs of operations that undo each other.
• A number plus its opposite makes 0: 6 + (−6) = 0
• A number times its reciprocal makes 1, when the number is not zero: 4 × 1/4 = 1

What to know as a beginner

It helps to remember that these laws do not all work in the same way for every operation. For example, addition and multiplication are commutative, but subtraction and division are not.

• 8 − 3 is not the same as 3 − 8
• 12 ÷ 3 is not the same as 3 ÷ 12

This is why understanding the rule matters more than memorizing a formula. The laws explain which changes are safe and which ones change the meaning of the calculation.

Practical tips

Use arithmetic laws to simplify problems step by step. If a sum looks awkward, you can often reorder or regroup numbers to make it easier. If a multiplication problem has brackets, the distributive law can help you break it into smaller parts.

When you solve a problem, check which law applies before rearranging anything. That habit helps you avoid mistakes and build confidence with numbers.