Laws of Arithmetic

Laws of arithmetic can be applied to expressions with numbers and/or variables. By applying laws of arithmetic, we can determine if algebraic expressions are equivalent without assigning values to the variables.

The Commutative Property. For expressions involving only addition or only multiplication, changing the order of the numbers will not change the result. 4 × 3 is equivalent to 3 × 4 on account of the Commutative Property of Multiplication. The Commutative Property does not hold for subtraction or division.

The Associative Property. When adding three numbers or multiplying three numbers, it does not matter whether you start with the first pair or the last. The same is true when either adding or multiplying four numbers, five numbers, etc.

The Associative Property often results in simpler mental computations. For instance, (14 + 6) + 7 + 2 is simpler to evaluate than 14 + (6 + (7 + 2)), although they both produce the same result of 29.

The Additive Inverse Property. Adding a number and its opposite always produces zero. For instance, 8 + - 8 = 0.

The Multiplicative Inverse Property. Multiplying a number and its reciprocal always produces 1. For instance, $$\displaystyle 8 \times \frac{1}{8} = 0$$.

The Identity Property. Multiplying or dividing an expression by 1 does not change its value; similarly, adding or subtracting 0 results in the original value. Due to the Identity Property, 5 + 9 produces the same result as (5 + 9) × 0 and (5 + 9) + 0.

The Distributive Property. Multiplying the sum of two addends by a number produces the same result as multiplying each addend by that number before finding the sum. In other words: a × (b + c) = ab + ac. For instance:

Applying the Distributive Property often results in simpler computations that can be completed using mental math.