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Chapter 1—Whole Numbers

1-2 Naming Numbers

 When you complete the work for this section, you should be able to: Identify the place value for any digit in any whole number. Demonstrate how periods of numbers are to be separated by a comma. Demonstrate the importance of using zero as a place keeper.

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.

Example 1

Consider the number 55. The first digit on the right is in the ones place. The actual value of that number is equal to that number, itself:  5.

Place values for the number 55

The second digit, however, is in the tens place. This means that its actual value is ten times the basic numerical value: 50.

The overall value of the number 55 is fifty plus five, or fifty five.

Example 2

This example uses a larger number, 384. Reading from right to left (¬), you can see that there is a 4 in the ones place, an 8 in the tens place, and a 3 in the hundreds place. So the value of this number is three-hundred eighty four:

Place values for the number 384

3 times a hundred
8 times ten
4 times one

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.

Place values for whole numbers, ones through billions

 Place Range of Values Ones 1 to 9 Tens 10 to 99 Hundreds 100 to 999 Thousands 1,000 to 9,999 Ten-Thousands 10,000 to 99,999 Hundred-Thousands 100,000 to 999,999 Millions 1,000,000 to 9,999,999 Ten-Millions 10,000,000 to 99,999,999 Hundred-Millions 100,000,000 to 999,999,999 Billions 1,000,000,000 to 9,999,999,999

Example 3

Here is an arrangement of place values for three different numbers: 268 and 579 924 and 213 890 756. You can also see how these three numbers are broken down into place values and how you can express the values in words.

 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 ninety thousand, seven-hundred fifty six

Periods of Numbers

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.

Place values can be divided into periods of three places apiece.

Place values are divided by units, thousands, millions, and so on. Sometimes people will separate the groups of with spaces, but you should use commas unless instructed otherwise.

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.

More Examples and Exercises

Use these interactive examples and exercises to strengthen your understanding and build your skills:

Periods of Place Values

Zero as a Place Keeper

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

For example, compare whole numbers 1 and 10. In the number ten, the zero 10 is a place keeper. 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 place keepers 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 place keeping zeros in the whole number system.

Technically speaking, place keeping 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.

Example 4

Insert a single place-keeping zero between the two highlighted numbers.

1.  1 85   ® 1805    Note: 185 becomes 1805

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