Basic Math and Pre-Algebra For Dummies (5 page)

You can use these simple up and down rules repeatedly to solve a longer string of added and subtracted numbers. For example, 3 + 1 − 2 + 4 − 3 − 2 means 3,
up
1,
down
2,
up
4,
down
3, and
down
2. In this case, the number line shows you that 3 + 1 − 2 + 4 − 3 − 2 = 1.

I discuss addition and subtraction in greater detail in Chapter
3
.

An important addition to the number line is the number 0, which means
nothing, zilch, nada.
Step back a moment and consider the bizarre concept of nothing. For one thing — as more than one philosopher has pointed out — by definition,
nothing
doesn't exist! Yet we routinely label it with the number 0, as in Figure 
1-8
.

 Actually, mathematicians have an even more precise labeling of
nothing
than zero. It's called the
empty
set, which is sort of the mathematical version of a box containing nothing. I introduce this concept, plus a little basic set theory, in Chapter
20
.

Nothing
sure is a heavy trip to lay on little kids, but they don't seem to mind. They understand quickly that when you have three toy trucks and someone
else takes away all three of them, you're left with zero trucks. That is, 3 − 3 = 0. Or, placing this on the number line, 3 − 3 means start at 3 and go down 3, as in Figure 
1-9
.

In Chapter
2
, I show you the importance of 0 as a
placeholder
in numbers and discuss how you can attach
leading zeros
to a number without changing its value.

When people first find out about subtraction, they often hear that you can't take away more than you have. For example, if you have four pencils, you can take away one, two, three, or even all four of them, but you can't take away more than that.

It isn't long, though, before you find out what any credit card holder knows only too well: You can, indeed, take away more than you have — the result is a
negative number.
For example, if you have $4 and you owe your friend $7, you're $3 in debt. That is, 4 − 7 = −3. The minus sign in front of the 3 means that the number of dollars you have is three less than 0. Figure 
1-10
shows how you place negative whole numbers on the number line.

Adding and subtracting on the number line works pretty much the same with negative numbers as with positive numbers. For example, Figure 
1-11
shows how to subtract 4 − 7 on the number line.

You find out all about working with negative numbers in Chapter
4
.

 Placing 0 and the negative counting numbers on the number line expands the set of counting numbers to the set of
integers
. I discuss the integers in further detail later in this chapter.

Suppose you start at 0 and circle every other number on a number line, as in Figure 
1-12
. As you can see, all the even numbers are now circled. In other words, you've circled all the
multiples of two.
(You can find out more about multiples in Chapter
8
.) You can now use this number line to multiply any number by two. For example, suppose you want to multiply 5 × 2. Just start at 0 and jump five circled spaces to the right.

This number line shows you that 5 × 2 = 10.

Similarly, to multiply −3 × 2, start at 0 and jump three circled spaces to the left (that is, in the negative direction). Figure 
1-13
shows you that −3 × 2 = −6. What's more, you can now see why multiplying a negative number by a positive number always gives you a negative result. (I talk about multiplying by negative numbers in Chapter
4
.)