Basic Math and Pre-Algebra For Dummies (7 page)

The counting numbers are infinite, which means they go on forever.

When you add two counting numbers, the answer is always another counting number. Similarly, when you multiply two counting numbers, the answer is always a counting number. Another way of saying this is that the set of counting numbers is
closed
under both addition and multiplication.

The set of
integers
arises when you try to subtract a larger number from a smaller one. For example, 4 − 6 = −2. The set of integers includes the following:

• The counting numbers
  
• Zero
  
• The negative counting numbers
  

Here's a partial list of the integers:

... −4 −3 −2 −1 0 1 2 3 4 ...

Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction.

Here's the set of
rational numbers:

• Integers
  
  • Counting numbers
    
  • Zero
    
  • Negative counting numbers
    
• Fractions
  

Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is that the rational numbers are closed under division.

Even if you filled in all the rational numbers, you'd still have points left unlabeled on the number line. These points are the irrational numbers.

An
irrational number
is a number that's neither a whole number nor a fraction. In fact, an irrational number can only be approximated as a
non-repeating decimal.
In other words, no matter how many decimal places you write down, you can always write down more; furthermore, the digits in this decimal never become repetitive or fall into any pattern. (For more on repeating decimals, see Chapter
11
.)

The most famous irrational number is π (you find out more about π when I discuss the geometry of circles in Chapter
17
):

Together, the rational and irrational numbers make up the
real numbers,
which comprise every point on the number line. In this book, I don't spend too much time on irrational numbers, but just remember that they're there for future reference.

Chapter 2

In This Chapter

Understanding how place value turns digits into numbers

Distinguishing whether zeros are important placeholders or meaningless leading zeros

Reading and writing long numbers

Understanding how to round numbers and estimate values

When you're counting, ten seems to be a natural stopping point — a nice, round number. The fact that our ten fingers match up so nicely with numbers may seem like a happy accident. But of course, it's no accident at all. Fingers were the first calculator that humans possessed. Our number system — Hindu-Arabic numbers — is based on the number ten because humans have 10 fingers instead of 8 or 12. In fact, the very word
digit
has two meanings: numerical symbol and finger.

In this chapter, I show you how place value turns digits into numbers. I also show you when 0 is an important placeholder in a number and why leading zeros don't change the value of a number. And I show you how to read and write long numbers. After that, I discuss two important skills: rounding numbers and estimating values.

The number system you're most familiar with — Hindu-Arabic numbers — has ten familiar digits:

• 0 1 2 3 4 5 6 7 8 9
  

Yet with only ten digits, you can express numbers as high as you care to go. In this section, I show you how it happens.

The ten digits in our number system allow you to count from 0 to 9. All higher numbers are produced using place value. Place value assigns a digit a greater or lesser value, depending on where it appears in a number. Each place in a number is ten times greater than the place to its immediate right.

To understand how a whole number gets its value, suppose you write the number 45,019 all the way to the right in Table 
2-1
, one digit per cell, and add up the numbers you get.

You have 4 ten thousands, 5 thousands, 0 hundreds, 1 ten, and 9 ones. The chart shows you that the number breaks down as follows:

• 45,019 = 40,000 + 5,000 + 0 + 10 + 9
  

In this example, notice that the presence of the digit 0 in the hundreds place means that zero hundreds are added to the number.

Although the digit 0 adds no value to a number, it acts as a placeholder to keep the other digits in their proper places. For example, the number 5,001,000 breaks down into 5,000,000 + 1,000. Suppose, however, you decide to leave all the 0s out of the chart. Table 
2-2
shows what you'd get.

The chart tells you that 5,001,000 = 50 + 1. Clearly, this answer is wrong!

 As a rule, when a 0 appears to the right of at least one digit other than 0, it's a placeholder. Placeholding zeros are important — always include them when you write a number. However, when a 0 appears to the left of every digit other than 0, it's a leading zero. Leading zeros serve no purpose in a number, so dropping them is customary. For example, place the number 003,040,070 on the chart (see Table 
2-3
).

The first two 0s in the number are leading zeros because they appear to the left of the 3. You can drop these 0s from the number, leaving you with 3,040,070. The remaining 0s are all to the right of the 3, so they're placeholders — be sure to write them in.

When you write a long number, you use commas to separate groups of three numbers. For example, here's about as long a number as you'll ever see:

• 234,845,021,349,230,467,304
  

Table 
2-4
shows a larger version of the place-value chart.

This version of the chart helps you read the number. Begin all the way to the left and read, “Two hundred thirty-four quintillion, eight hundred forty-five quadrillion, twenty-one trillion, three hundred forty-nine billion, two hundred thirty million, four hundred sixty-seven thousand, three hundred four.”

 When you read and write whole numbers, don't say the word
and
. In math, the word
and
means you have a decimal point. That's why, when you write a check, you save the word
and
for the number of cents, which is usually expressed as a decimal or sometimes as a fraction. (I discuss decimals in Chapter
11
.)