Basic Math and Pre-Algebra For Dummies (6 page)

You can also use the number line to divide. For example, suppose you want to divide 6 by some other number. First, draw a number line that begins at 0 and ends at 6, as in Figure 
1-14
.

Now, to find the answer to 6 ÷ 2, just split this number line into two equal parts, as in Figure 
1-15
. This split (or
division
) occurs at 3, showing you that 6 ÷ 2 = 3.

Similarly, to divide 6 ÷ 3, split the same number line into three equal parts, as in Figure 
1-16
. This time you have two splits, so use the one closest to 0. This number line shows you that 6 ÷ 3 = 2.

But suppose you want to use the number line to divide a small number by a larger number. For example, maybe you want to know the answer to 3 ÷ 4. Following the method I show you earlier, first draw a number line from 0 to 3. Then split it into four equal parts. Unfortunately, none of these splits has landed on a number. It's not a mistake — you just have to add some new numbers to the number line, as you can see in Figure 
1-17
.

Welcome to the world of
fractions.
With the number line labeled properly, you can see that the split closest to 0 is
. This image tells you that
. The similarity of the expression 3 ÷ 4 and the fraction
is no accident. Division and fractions are closely related. When you divide, you cut things up into equal parts, and fractions are often the result of this process. (I explain the connection between division and fractions in more detail in Chapters
9
and
10
.)

Fractions help you fill in a lot of the spaces on the number line that fall between the counting numbers. For example, Figure 
1-18
shows a close-up of a number line from 0 to 1.

This number line may remind you of a ruler or a tape measure, with a lot of tiny fractions filled in. In fact, rulers and tape measures really are portable number lines that allow carpenters, engineers, and savvy do-it-yourselfers to measure the length of objects with precision.

Adding fractions to the number line expands the set of integers to the set of
rational numbers.
I discuss the rational numbers in greater detail in Chapter
25
.

 In fact, no matter how small things get in the real world, you can always find a tiny fraction to approximate it as closely as you need. Between any two fractions on the number line, you can always find another fraction. Mathematicians call this trait the
density
of fractions on the real number line, and this type of density is a topic in a very advanced area of math called
real analysis.

In the preceding section, you see how the number line grows in both the positive and negative directions and fills in with a lot of numbers in between. In this section, I provide a quick tour of how numbers fit together as a set of nested systems, one inside the other.

When I talk about a set of numbers, I'm really just talking about a group of numbers. You can use the number line to deal with four important sets of numbers:

• Counting numbers (also called natural numbers):
  The set of numbers beginning 1, 2, 3, 4 ... and going on infinitely
  
• Integers:
  The set of counting numbers, zero, and negative counting numbers
  
• Rational numbers:
  The set of integers and fractions
  
• Real numbers:
  The set of rational and irrational numbers
  

The sets of counting numbers, integers, rational, and real numbers are nested, one inside another. This nesting of one set inside another is similar to the way that a city (for example, Boston) is inside a state (Massachusetts), which is inside a country (the United States), which is inside a continent (North America). The set of counting numbers is inside the set of integers, which is inside the set of rational numbers, which is inside the set of real numbers.

The set of
counting numbers
is the set of numbers you first count with, starting with 1. Because they seem to arise naturally from observing the world, they're also called the
natural numbers:

1  2   3    4    5    6   7    8    9 ...