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Chapter 1—Whole Numbers

1-1 Introducing Whole Numbers

Introducing Whole Numbers

Natural Numbers

When you count on your fingers—one-two-three-four, and so on—you are counting with natural numbers.  Using your fingers and thumbs for counting, you can count from 1 to 10. If you include all your toes, you can count another ten whole numbers. This is the kind of counting that comes most naturally.

Of course we need to count a lot higher than ten. We need to count into the tens. hundreds, thousands, millions, and so on. So the natural number system goes far beyond the limitations of simple finger-and-toe counting. It begins with the number 1 and goes upward as far as we can imagine. And even then, it keeps going. We can show the natural number system this way:

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

where the ellipsis (three dots in row) indicates that the counting continues without end.

Another way to portray the natural number system is with a number line. A number line is a scale that looks and works much like a measuring stick. You can see that the numbers are marked on the scale and arranged in order, from left to right. The dashes and arrow ( ) at the right end of the line tells us that the scale and the counting can go on this way forever. Natural number line.

Whole Numbers

Although the natural number system proved to be very handy for conducting business in ancient times, it lacked one very important feature: the concept of zero. When a zero is added at the beginning of the natural number system, the system becomes the whole number system:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

The whole number system begins with zero and counts upward through tens, hundreds, thousands, millions, and so on.

The number line for the whole number system looks like this: Whole number line

The scale for the whole number line begins with zero and runs to the right. How far does it run to the right? We can say that the whole number system extends from 0 to ¥ (spoken as "from zero to infinity").

The Decimal Numbering System

The whole-number system uses only ten characters--0 through 9. They are the characters of our familiar decimal numbering system. Note that the deci- in decimal means ten -- the total number of digits (fingers and thumbs) on our two hands.

 0 1 2 3 4 5 6 7 8 9

Every number we might ever want to express can be written as a combination of these ten, simple digits.

The Value of a Decimal Number

Numbers have a certain value, or magnitude. In our decimal numbering system, the numeral 6 represents six things (stones, fingers, sticks, etc.). The numeral 4, on the other hand, represents four such things. Thus we can say that 6 is larger than 4. Likewise, we can say that 9 has a greater value than 2, 3 has a smaller value than 5, and 9 is the largest of all the decimal digits. The value, or magnitude, of a decimal number can also be indicated on a number line: You can see that the value of these decimal whole numbers increase from left to right. (Of course you can also say that the values decrease from right to left.) Values increase from left to right
on the whole-number line

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