Basic Math and Pre-Algebra For Dummies (4 page)

Square numbers are also a great first step on the way to understanding exponents, which I introduce later in this chapter and explain in more detail in Chapter
4
.

Some numbers can be placed in rectangular patterns. Mathematicians probably should call numbers like these “rectangular numbers,” but instead they chose the term
composite numbers.
For example, 12 is a composite number because you can place 12 objects in rectangles of two different shapes, as in Figure 
1-2
.

As with square numbers, arranging numbers in visual patterns like this tells you something about how multiplication works. In this case, by counting the sides of both rectangles, you find out the following:

3 × 4 = 12

2 × 6 = 12

Similarly, other numbers such as 8 and 15 can also be arranged in rectangles, as in Figure 
1-3
.

As you can see, both these numbers are quite happy being placed in boxes with at least two rows and two columns. And these visual patterns show this:

2 × 4 = 8

3 × 5 = 15

The word
composite
means that these numbers are
composed of
smaller numbers. For example, the number 15 is composed of 3 and 5 — that is, when you multiply these two smaller numbers, you get 15. Here are all the composite numbers from 1 to 16:

4    6    8    9  10  12  14  15  16

Notice that all the square numbers (see “Getting square with square numbers”) also count as composite numbers because you can arrange them in boxes with at least two rows and two columns. Additionally, a lot of other nonsquare numbers are also composite numbers.

Some numbers are stubborn. Like certain people you may know, these numbers — called
prime numbers
— resist being placed in any sort of a box. Look at how Figure 
1-4
depicts the number 13, for example.

Try as you may, you just can't make a rectangle out of 13 objects. (That fact may be one reason the number 13 got a bad reputation as unlucky.) Here are all the prime numbers less than 20:

2    3    5   7  11   13  17  19

As you can see, the list of prime numbers fills the gaps left by the composite numbers (see the preceding section). Therefore, every counting number is either prime or composite. The only exception is the number 1, which is neither prime nor composite. In Chapter
8
, I give you a lot more information about prime numbers and show you how to
decompose
a number — that is, break down a composite number into its prime factors.

Here's an old question whose answer may surprise you: Suppose you took a job that paid you just 1 penny the first day, 2 pennies the second day, 4 pennies the third day, and so on, doubling the amount every day, like this:

1  2    4    8  16    32   64  128    256    512 ...

As you can see, in the first ten days of work, you would've earned a little more than $10 (actually, $10.23 — but who's counting?). How much would you earn in 30 days? Your answer may well be, “I wouldn't take a lousy job like that in the first place.” At first glance, this looks like a good answer, but here's a glimpse at your second ten days' earnings:

... 1,024   2,048   4,096   8,192  16,384    32,768   65,536   131,072     262,144   524,288 ...

By the end of the second 10 days, your total earnings would be over $10,000. And by the end of 30 days, your earnings would top out around $10,000,000! How does this happen? Through the magic of exponents (also called
powers
). Each new number in the sequence is obtained by multiplying the previous number by 2:

As you can see, the notation 2
⁴
means
multiply 2 by itself 4 times.

You can use exponents on numbers other than 2. Here's another sequence you may be familiar with:

1  10   100   1,000   10,000   100,000   1,000,000 …

In this sequence, every number is 10 times greater than the number before it. You can also generate these numbers using exponents:

This sequence is important for defining
place value,
the basis of the decimal number system, which I discuss in Chapter
2
. It also shows up when I discuss decimals in Chapter
11
and scientific notation in Chapter
15
. You find out more about exponents in Chapter
5
.

As kids outgrow counting on their fingers (and use them only when trying to remember the names of all seven dwarfs), teachers often substitute a picture of the first ten numbers in order, like the one in Figure 
1-5
.

This way of organizing numbers is called the
number line.
People often see their first number line — usually made of brightly colored construction paper — pasted above the blackboard in school. The basic number line provides a visual image of the
counting numbers
(also called the
natural numbers
), the numbers greater than 0. You can use it to show how numbers get bigger in one direction and smaller in the other.

In this section, I show you how to use the number line to understand a few basic but important ideas about numbers.

You can use the number line to demonstrate simple addition and subtraction. These first steps in math become a lot more concrete with a visual aid. Here's the main point to remember:

• As you go
  right
  , the numbers go
  up,
  which is
  addition
  (+).
  
• As you go
  left
  , the numbers go
  down,
  which is
  subtraction
  (−).
  

For example, 2 + 3 means you
start at
2 and
jump up
3 spaces to 5, as Figure 
1-6
illustrates.

As another example, 6 − 4 means
start
at 6 and
jump down
4 spaces to 2. That is, 6 − 4 = 2. See Figure 
1-7
.