Basic Math and Pre-Algebra For Dummies (15 page)

Chapter 4

In This Chapter

Identifying which operations are inverses of each other

Knowing the operations that are commutative, associative, and distributive

Performing the Big Four operations on negative numbers

Using four symbols for inequality

Understanding exponents, roots, and absolute values

When you understand the Big Four operations that I cover in Chapter
3
— adding, subtracting, multiplying, and dividing — you can begin to look at math on a whole new level. In this chapter, you extend your understanding of the Big Four operations and move beyond them. I begin by focusing on four important properties of the Big Four operations: inverse operations, commutative operations, associative operations, and distribution. Then I show you how to perform the Big Four on negative numbers.

I continue by introducing you to some important symbols for inequality. Finally, you're ready to move beyond the Big Four by discovering three more advanced operations: exponents (also called
powers
), square roots (also called
radicals
), and absolute values.

When you know how to do the Big Four operations — add, subtract, multiply, and divide — you're ready to grasp a few important
properties
of these
important operations. Properties are features of the Big Four operations that always apply, no matter what numbers you're working with.

In this section, I introduce you to four important ideas: inverse operations, commutative operations, associative operations, and the distributive property. Understanding these properties can show you hidden connections among the Big Four operations, save you time when calculating, and get you comfortable working with more-abstract concepts in math.

Each of the Big Four operations has an
inverse
— an operation that undoes it. Addition and subtraction are inverse operations because addition undoes subtraction, and vice versa. For example, here are two equations with inverse operations:

• 1 + 2 = 3
  
• 3 − 2 = 1
  

In the first equation, you start with 1 and add 2 to it, which gives you 3. In the second equation, you have 3 and take away 2 from it, which brings you back to 1. The main idea here is that you're given a starting number — in this case, 1 — and when you add a number and then subtract the same number, you end up again with the starting number. This shows you that subtraction undoes addition.

Similarly, addition undoes subtraction — that is, if you subtract a number and then add the same number, you end up where you started. For example,

• 184 − 10 = 174
  
• 174 + 10 = 184
  

This time, in the first equation, you start with 184 and take away 10 from it, which gives you 174. In the second equation, you have 174 and add 10 to it, which brings you back to 184. In this case, starting with the number 184, when you subtract a number and then add the same number, the addition undoes the subtraction and you end up back at 184.

In the same way, multiplication and division are inverse operations. For example,

This time, you start with the number 4 and multiply it by 5 to get 20. And then you divide 20 by 5 to return to where you started at 4. So division undoes multiplication. Similarly,

Here, you start with 30, divide by 10, and multiply by 10 to end up back at 30. This shows you that multiplication undoes division.

Addition and multiplication are both commutative operations.
Commutative
means that you can switch around the order of the numbers without changing the result. This property of addition and multiplication is called the
commutative property.
Here's an example of how addition is commutative:

• 3 + 5 = 8   is the same as   5 + 3 = 8
  

If you start out with 5 books and add 3 books, the result is the same as if you start with 3 books and add 5. In each case, you end up with 8 books.

And here's an example of how multiplication is commutative:

If you have 2 children and want to give them each 7 flowers, you need to buy the same number of flowers as someone who has 7 children and wants to give them each 2 flowers. In both cases, someone buys 14 flowers.

 In contrast, subtraction and division are
non-commutative
operations. When you switch the order of the numbers, the result changes.

Here's an example of how subtraction is non-commutative: