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Chapter 1    Whole Numbers

1-2    Naming Numbers

The whole-number system consists of only ten different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.  Although there are only ten digits, we can combine any number of them to count from zero to infinity ( 0 to  � ). In order to create so many combinations of numbers, it is necessary to agree upon some carefully crafted rules.

The rules for creating an infinite number of values out of ten simple digits are based upon the idea of place values. The value of a digit depends upon its place in a series of digits.

What comes after the hundreds place? The thousands place. And after that? The ten-thousands place, the hundred-thousands place, and the millions place. Table 4-1 shows the place values up to a billion. This is far enough to account for whole numbers ranging from 0 to 9,999,999,999. We could keep going a lot farther than a billion, but this is far enough for the work in this course.

Table 1-1.  Place values for whole numbers, ones through billions

Examples

Figure 1-4 shows the arrangement of place values for three different numbers: 268 and 575 921 and 213 890 756. You can also see how those three numbers are broken down into place values and read their customary spoken forms.

Example Place Values Customary Spoken Form 268 8 ones 6 tens 2 hundreds Two hundred, sixty, eight 579 924 4 ones 2 tens 9 hundreds 9 thousands 7 ten-thousands 5 hundred-thousands Five-hundred seventy-nine thousand, nine nundred and twenty four 213 890 756 6 ones 5 tens 7 hundreds 0 thousands 9 ten-thousands 8 hundred-thousands 3 millions 1 ten-millions 2 hundred-millions Two-hundred thirteen million, eight-hundred ninty thousand, seven-hundred fifty six

Figure 1-4. Examples of place values.

Exercises

Indicate the place value for the highlighted digit.
Click the  ?  symbol to see the correct answer.

Periods

Even after you have mastered the principles of place values for whole numbers, it still takes some effort to sort out the meaning of numbers larger than ten thousand or so. Consider the number 379092361. It takes some time to figure out where the place values begin. But if we divide the number into periods of three digits apiece, it all becomes much simpler to understand:  379,092,361.

Figure 1-5 shows how the place values are divided:   units, thousands, millions, and so on. Sometimes people will separate the groups of with spaces, but you should use commas unless instructed otherwise.

Figure 1-5. Place values can be divided into periods of three places apiece.

Using commas to separate the periods, we can conveniently read 379,092,361 as: 379 million (the millions period) 92 thousand (the thousands period) 361 (the units period) Note When a number is in the thousands, between 1000 and 9999, you do not have to use a comma to separate the thousands digit from the units phase. For example, you may write 1,256 or 1256. But once the values get into the tens-of-thousands and beyond, you should use the comma to separate the thousands phase from the units phase. For example, you should write 12,354 in order to make it easier to understand the value of 12354.

Exercises

Rewrite these numbers, inserting commas to separate the periods of place values.
Click the  ?  symbol to see the correct answer.

Zero as a Placekeeper

The number zero (0) is does not have any value in the whole number system. However, zero plays a very important role as a placekeeper.

For example, compare whole numbers 1 and 10. In the number ten, the zero 10 is a placekeeper. This is a vital feature because, if you forget to use the zero, the ten looks like a one.  Try it yourself:  Write the number 10 on a piece of paper, then erase the 0. Now its a just a 1.

Suppose someone is instructed to give you an envelope with $5000 cash in it. This person, however, is careless about placekeepers and ignores all three of them. How much will be in your envelope? Just $5. Five dollars is nice, but it is way, way short of $5000. Again, you can appreciate the importance of the placekeeping zeros in the whole number system.

Technically speaking, placekeeping zeros shift the place value of all numbers to its left. In the number 321, for example the 3 is in the hundreds place and the 2 is in the tens place. If you insert a zero between the 2 and the 1, the result is 3201. The zero shifts the 3 to the thousands place and the 2 to the hundreds place. Indeed, inserting this zero caused all numbers to its left to be shifted upward one place value.

Inserting a pair of zeros causes all numbers to the left to shift two place values. Begin with 321 again, an insert two zeros between the 2 and the 1. The result is 32001—a significantly larger number than 321.

Notes:

• Inserting a zero at the right end of a whole number shifts all the others upward one place value. The result is exactly ten times larger than before the zero is added.
• Inserting a zero at the left end of a whole number does not affect its value at all. Zeros that are used at the left end of a number are called leading zeros, and are used only for special reasons.

Examples

Insert a single placekeeper zero between the two highlighted numbers.

1.  1 85   � 1805    Note: 185 becomes 1805

2.  3341   � 33,401      Note: 3341 becomes 33,401

Copyright � 2006, David L. Heiserman
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