Early in a first year algebra course, students will learn about real numbers and their properties. The properties may be very confusing to distinguish at first, but there are some specific things to look for to be able to quickly identify an equation with its corresponding property.
Here is the list of properties of real numbers with their explanations.
- The commutative property of addition states that changing the order when adding doesn’t affect the sum. Think of the word commute and how one might commute to work or to school. This means that a person travels to and from work or school. The distance is the same both directions, assuming the same route is taken to and from. So for any two numbers, a and b, a + b = b + a demonstrates the commutative property
- The commutative property of multiplication states that changing the order when multiplying doesn’t affect the outcome. For any two numbers a and b, a ・ b = b ・ a.
- The associative property of addition states that changing the grouping does not affect the outcome when adding. Think of the meaning of the word associate. When one associates with someone, he or she is grouped with that person. Same can be applied here. For any numbers a,b and c, a +(b + c) = (a + b)+c.
- The associative property of multiplication follows the same principle and states that changing the grouping does not affect the outcome when multiplying. Therefore, (ab)c= a(bc).
- The distributive property of multiplication shows how multiplication distributes over addition. For numbers a,b and c, a ・ (b + c) = ab + ac.
- The identity properties show how when you add 0 to a number or multiply 1 to a number you still get that number. Identity property of addition is (a + 0 = a) and the Identity property of multiplication is (a∙1= a).
- The inverse property of addition shows how when you add a number to its inverse (or opposite), the result is 0. For example -4 + 4= 0.
- The inverse property of multiplication shows how when you multiply a number by its inverse, the result is 1. For example, 2(1/2) = 1.
Here are some examples which will help you identify which property is being used.
Commutative property of addition
1 + 2 = 2 + 1
3 = 3
Commutative property of multiplication
2 ・ 3 = 3 ・ 2
6 = 6
It doesn’t matter what order we add or multiply numbers together, the result will be the same.
Associative property of addition
1 + (2 + 3) = (1 + 2) + 3
1 + 5 = 3 + 3 (from the order of operations, we add what’s inside the parentheses first)
6 = 6
Associative property of multiplication
2 ・ (3 ・ 4) = (2 ・ 3) ・ 4
2 ・ 12 = 6 ・ 4 (from the order of operations, we multiply what’s inside the parentheses first)
24 = 24
Grouping the numbers differently does not affect the answer when adding or multiplying numbers.
Distributive property of multiplication
2(4 + 5) = 2(4) + 2(5) (multiplied the 2 by 4 and then 2 by 5)
2(9) = 2(4) + 2(5) (added numbers inside the parentheses on left side of the = sign)
18 = 8 + 10 (multiplied on both sides of the equation)
18 = 18
Identity property of addition and identity property of multiplication
2 + 0 = 2, 2(1) = 2
When you add 0 to a number, the result is that number. When you multiply 1 to a number, the result is that number.
Inverse property of addition and inverse property of multiplication
2 + (-2) = 0, 2(1/2) = 1
When you add a number to its opposite (the negative of that number), the result is 0. When you multiply a number by its inverse (switch numerator and denominator, ex: 1/3 is inverse of 3/1), the result is 1.
I use this quick guide when assisting my students that I tutor and believe it will help anyone with trouble identifying and applying the properties of real numbers.