Operations

The Commutative Property states that for any numbers a and b, the following is always true:

For example, 3 + 5 = 5 + 3. We can see that this is true because 3 + 5 = 8 and 5 + 3 = 8, so 3 + 5 and 5 + 3 are equal to each other. Another way to think of the commutative property is the following: if you have a quarter and a dime in your pocket, and you add them together, you will come up with the same amount of money whether you add the quarter to the dime or the dime to the quarter.

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:

This new expression is easier to evaluate, because 48 + 52 = 100 and 100 + 33 + 17 = 150. It is easier to add numbers to numbers which end in "0". This expression can be made even easier to evaluate with the associative property:

The Associative Property states that for any numbers a, b, and c, the following is always true:

For example, (2 + 4) + 7 = 2 + (4 + 7). We can see that this is true because (2 + 4) + 7 = 6 + 7 = 13 and 2 + (4 + 7) = 2 + 11 = 13, so (2 + 4) + 7 and 2 + (4 + 7) are equal to each other. Or we can once again think about it using the example of coins: if I have a nickel and a dime in my left pocket and a quarter in my right pocket, I will have the same amount of money if I take the dime out of my left pocket and put it in my right pocket with the quarter.

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 pairs:

It's a lot easier to add these three numbers in one's head than to add the original five numbers one by one, and both methods yield the same answer--161.

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.

One final property of addition that will be very useful in algebra is the Identity Property, which says that for any number a, the following are always true:

The Identity Property of Addition says that a number does not change its identity when 0 is added. For example, 12 + 0 = 12. 0 + 17 = 17. Or, if someone is given zero dollars, the amount of money he has does not change.

These properties can be used in any order. Right now, they are useful because they make it easier to add long strings of numbers. Later, they will help us to solve algebraic equations, which we will discuss in Inverse Operations.

Here are some examples to show how these properties can make mental math easier:

Example 1. 12 + 67 + 98 = ?
Commutative Property: 12 + 67 + 98 = 12 + 98 + 67
12 + 98 + 67 = 110 + 67 = 177

Example 2. (13 + 21) + (9 + 5) + 5 = ?

Associative Property: (13 + 21) + (9 + 5) + 5 = 13 + (21 + 9) + (5 + 5)
13 + (21 + 9) + (5 + 5) = 13 + 30 + 10 = 53
Example 3. 54 + 17 + 6 + 12 + 3 + 18 = ?

Commutative Property: 54 + 17 + 6 + 12 + 3 + 18 = 54 + 6 + 17 + 3 + 12 + 18
Associative Property: 54 + 6 + 17 + 3 + 12 + 18 = (54 + 6) + (17 + 3) + (12 + 18)
(54 + 6) + (17 + 3) + (12 + 18) = 60 + 20 + 30 = 110