1.07 Number operation properties

Try calculating 29+38+12 in your head. Which two numbers did you add together first? If you added the 38 and 12 together first, you were using the associative law. The associative law lets us evaluate the operations in any order, so long as the operations are all addition or all multiplication. Mathematically, this looks like we are adding or moving brackets to change which pair of numbers we add or multiply first.

The associative law for addition

This law doesn't apply to subtraction or division.

For example: 22-7-5 \neq 22-(7-5) and 24 \div 6 \div 2 \neq 24 \div (6 \div 2).

Examples

Which of the following are true?

a) \,4+3=3+4 \quad \quadb) \,4-3=3-4 \quad \quadc) \,4\times 3=3\times 4 \quad \quadd) \,4\div 3=3\div 4

If you answered a) and c), you're correct. These two show applications of the commutative law which lets us swap the numbers on either side of the operation. Notice that since only a) and c) are true, the commutative law can be used for addition and multiplication, but not subtraction and division.

The commutative law for multiplication

Examples

This mental strategy is a similar to the commutative law but can be used when we have more than one of the same operation.

When calculating 13+18+7 which number would it be easier to add first, 18 or 7? Unless you are a calculator it is usually easier to add 7 from 13 first and then add 18 to their sum. This is an example of using reordering.

Reordering lets us rearrange the numbers in the expression to make the calculations easier when solving from left to right. The rules for using reordering are that all the operations must be same and the first number cannot be moved (notice that 13 stays as the first number in the expression), unless the operation is also commutative (i.e. if it is either addition or multiplication).

Examples

The distributive law applies whenever there is multiplication outside some brackets and addition inside the brackets, such as the expression 3 \times (5+2). We apply the multiplication to each term inside the brackets individually, and add the results together:

The distributive law for multiplication

This law will be very useful when we study algebra, but for now it is still a clever strategy for mental arithmetic.

Is there an easy way to solve 102 \times 13 There is. The trick is find an "easy to multiply by" number close to 102. In this case we can use 100. So instead of trying to calculate 102 \times 13 we can instead calculate (100+2) \times 13. We can then use the distributive law to expand the brackets to get (100 \times 13) + (2 \times 13), which is much easier to multiply and add.

The distributive law can be used whenever there are brackets with addition (or subtraction) inside, and multiplication on the outside (on either side). It can also be used when the brackets are being divided from the right side.

For example:

\begin{aligned} (24+6) \times 2 &= (24 \times 2)+ (6\times2) \\ 2 \times (24+6) &= (2\times 24) + (2 \times 6) \\ (24+6) \div 2 &= (24 \div 2) + (6 \div 2) \end{aligned}

We cannot apply the distributive law to division from the left side.

For example: 24 \div (6+2) \neq (24 \div 6) + (24 \div 2)

We can show the distributive law visually as well through area. We can either break up the total area into two simpler areas or subtract some excess area from an approximate total.

We can use a similar trick to make division questions like 168 \div 14 easier. What is a number close to 168 that is "easy to divide by 14"? One way we can break up 168 is into 140 and 28, splitting a difficult to divide number into two easy to divide numbers.

Examples