Chapter 1Whole 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.
Place values for whole numbers, ones through billions
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.
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 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.
More Examples and Exercises
Use these interactive examples and exercises to strengthen your understanding and build your skills:
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 32001a significantly larger number than 321.
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