1.8 The real numbers  (Page 3/13)

What do these examples tell us?

Every rational number can be written both as a ratio of integers , $(\frac{p}{q},$ where p and q are integers and $q\ne 0),$ and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Are there any decimals that do not stop or repeat? Yes!

The number $\pi$ (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

We can even create a decimal pattern that does not stop or repeat, such as

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number     s .

All the numbers we use in elementary algebra are real numbers. [link] illustrates how the number sets we’ve discussed in this section fit together.