Sometimes it is necessary to add long strings of numbers without a calculator. For example, one might be asked to find 48 + 33 + 52 + 11 + 17. This sum is difficult to compute without a calculator, but the task can be made a lot easier by knowing some simple properties of addition. In this section, we will focus on these properties, which will help make "mental math" easier and will be useful in later sections of Pre-Algebra.
The Commutative Property states that for any numbers a and b, the following is always true:
By the commutative property, if we add two or more numbers, we can always add them in any order. This is useful because it might be easier to add numbers in a different order than the order given. In our example above, it takes a long time to add the numbers from left to right (try it). However, because addition has the commutative property, we can switch the order of the numbers in the expression:
The Associative Property states that for any numbers a, b, and c, the following is always true:
Not only can we add numbers in any order, we can also add pairs of numbers
within the expression before adding them all together. In other words, we can
put parenthesis around any two (or more) numbers and add those numbers
separately. Using our example above, we can rearrange the numbers using the
commutative property and then use the associative property to add them in
The Commutative Property of Addition can be remembered by remembering that when only addition is involved, numbers can move ("commute") to anywhere in the expression. The Associative Property of Addition can be remembered by remembering that any numbers that are being added together can "associate" with each other. Another good rule of thumb is, when trying to decide which properties to use, look for numbers that add up to multiples of 10; these should be added first because they are easy to add to other numbers.