Basic Math and Pre-Algebra For Dummies (16 page)

Subtraction is non-commutative, so if you have $6 and spend $4, the result is
not
the same as if you have $4 and spend $6. In the first case, you still have $2 left over. In the second case, you
owe
$2. In other words, switching the numbers turns the result into a negative number. (I discuss negative numbers later in this chapter.)

And here's an example of how division is non-commutative:

For example, when you have five dog biscuits to divide between two dogs, each dog gets two biscuits and you have one biscuit left over. But when you switch the numbers and try to divide two biscuits among five dogs, you don't have enough biscuits to go around, so each dog gets none and you have two left over.

Addition and multiplication are both
associative operations
, which means that you can group them differently without changing the result. This property of addition and multiplication is also called the
associative property.
Here's an example of how addition is associative. Suppose you want to add 3 + 6 + 2. You can calculate in two ways:

In the first case, I start by adding 3 + 6 and then add 2. In the second case, I start by adding 6 + 2 and then add 3. Either way, the sum is 11.

And here's an example of how multiplication is associative. Suppose you want to multiply
. You can calculate in two ways:

In the first case, I start by multiplying 5 × 2 and then multiply by 4. In the second case, I start by multiplying 2 × 4 and then multiply by 5. Either way, the product is 40. In contrast, subtraction and division are non-associative operations. This means that grouping them in different ways changes the result.

 Don't confuse the commutative property with the associative property. The commutative property tells you that it's okay
to switch
two numbers that you're adding or multiplying. The associative property tells you that it's okay to
regroup
three (or more) numbers using parentheses.

Taken together, the commutative and associative properties allow you to completely rearrange and regroup a string of numbers that you're adding or multiplying without changing the result. You'll find the freedom to rearrange expressions as you like to be very useful as you move on to algebra in Part
V
.

If you've ever tried to carry a heavy bag of groceries, you may have found that distributing the contents into two smaller bags is helpful. This same concept also works for multiplication.

In math,
distribution
(also called the
distributive property of multiplication over addition
) allows you to split a large multiplication problem into two smaller ones and add the results to get the answer.

For example, suppose you want to multiply these two numbers:

• 17 × 101
  

You can go ahead and just multiply them, but distribution provides a different way to think about the problem that you may find easier. Because 101 = 100 + 1, you can split this problem into two easier problems, as follows:

You take the number outside the parentheses, multiply it by each number inside the parentheses one at a time, and then add the products. At this point, you may be able to calculate the two multiplications in your head and then add them up easily:

• = 1,700 + 17 = 1,717
  

Distribution becomes even more useful when you get to algebra in Part
V
.

In Chapter
1
, I show you how to use the number line to understand how negative numbers work. In this section, I give you a closer look at how
to perform the Big Four operations with negative numbers. Negative numbers result when you subtract a larger number from a smaller one. For example,

• 5 − 8 = −3
  

In real-world applications, negative numbers represent debt. For example, if you have only five chairs to sell but a customer pays for eight of them, you owe her three more chairs. Even though you may have trouble picturing −3 chairs, you still need to account for this debt, and negative numbers are the right tool for the job.

The great secret to adding and subtracting negative numbers is to turn every problem into a series of ups and downs on the number line. When you know how to do this, you find that all these problems are quite simple.

So in this section, I explain how to add and subtract negative numbers on the number line. Don't worry about memorizing every little bit of this procedure. Instead, just follow along so you get a sense of how negative numbers fit onto the number line. (If you need a quick refresher on how the number line works, see Chapter
1
.)

When you're adding and subtracting on the number line, starting with a negative number isn't much different from starting with a positive number. For example, suppose you want to calculate −3 + 4. Using the up and down rules, you start at −3 and go up 4:

So −3 + 4 = 1.

Similarly, suppose you want to calculate −2 − 5. Again, the up and down rules help you out. You're subtracting, so move to the left: start at −2, down 5: