# 1.8 The real numbers  (Page 3/13)

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$\text{These decimals either stop or repeat.}$

What do these examples tell us?

Every rational number can be written both as a ratio of integers , $\left(\frac{p}{q},$ where p and q are integers and $q\ne 0\right),$ 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:

Fractions Integers
Number $\frac{4}{5}$ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$ $\frac{13}{4}$ $-\phantom{\rule{0.2em}{0ex}}\frac{20}{3}$ $-2$ $-1$ $0$ $1$ $2$ $3$
Ratio of Integers $\frac{4}{5}$ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$ $\frac{13}{4}$ $-\phantom{\rule{0.2em}{0ex}}\frac{20}{3}$ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{1}$ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{1}$ $\frac{0}{1}$ $\frac{1}{1}$ $\frac{2}{1}$ $\frac{3}{1}$
Decimal Form $0.8$ $-0.875$ $3.25$ $-6.\stackrel{\text{–}}{6}$ $-2.0$ $-1.0$ $0.0$ $1.0$ $2.0$ $3.0$

## Rational number

A rational number is a number of the form $\frac{p}{q},$ where p and q are integers and $q\ne 0.$

Its decimal form stops or repeats.

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.

$\pi =3.141592654...$

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

$2.01001000100001\dots$

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

## Irrational number

An irrational number    is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

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

## Rational or irrational?

If the decimal form of a number

• repeats or stops , the number is rational .
• does not repeat and does not stop , the number is irrational .

Given the numbers $0.58\stackrel{\text{–}}{3},0.47,3.605551275...$ list the rational numbers irrational numbers.

## Solution

$\begin{array}{cccccc}\text{Look for decimals that repeat or stop.}\hfill & & & & & \text{The}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{repeats in}\phantom{\rule{0.2em}{0ex}}0.58\stackrel{\text{–}}{3}.\hfill \\ & & & & & \text{The decimal}\phantom{\rule{0.2em}{0ex}}0.47\phantom{\rule{0.2em}{0ex}}\text{stops after the}\phantom{\rule{0.2em}{0ex}}7.\hfill \\ & & & & & \text{So}\phantom{\rule{0.2em}{0ex}}0.58\stackrel{\text{–}}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0.47\phantom{\rule{0.2em}{0ex}}\text{are rational.}\hfill \end{array}$

$\begin{array}{cccccc}\text{Look for decimals that neither stop nor repeat.}\hfill & & & & & 3.605551275\text{…}\phantom{\rule{0.2em}{0ex}}\text{has no repeating block of}\hfill \\ & & & & & \text{digits and it does not stop.}\hfill \\ & & & & & \text{So}\phantom{\rule{0.2em}{0ex}}3.605551275\text{…}\phantom{\rule{0.2em}{0ex}}\text{is irrational.}\hfill \end{array}$

For the given numbers list the rational numbers irrational numbers: $0.29,0.81\stackrel{\text{–}}{6},2.515115111\text{…}.$

$0.29,0.81\stackrel{\text{–}}{6}$ $2.515115111\text{…}$

For the given numbers list the rational numbers irrational numbers: $2.6\stackrel{\text{–}}{3},0.125,0.418302\text{…}$

$2.6\stackrel{\text{–}}{3},0.125$ $0.418302\text{…}$

For each number given, identify whether it is rational or irrational: $\sqrt{36}$ $\sqrt{44}.$

1. Recognize that 36 is a perfect square, since ${6}^{2}=36.$ So $\sqrt{36}=6,$ therefore $\sqrt{36}$ is rational.
2. Remember that ${6}^{2}=36$ and ${7}^{2}=49,$ so 44 is not a perfect square. Therefore, the decimal form of $\sqrt{44}$ will never repeat and never stop, so $\sqrt{44}$ is irrational.

For each number given, identify whether it is rational or irrational: $\sqrt{81}$ $\sqrt{17}.$

rational irrational

For each number given, identify whether it is rational or irrational: $\sqrt{116}$ $\sqrt{121}.$

irrational rational

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 .

## Real number

A real number is a number that is either rational or irrational.

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.

What is the lcm of 340
How many numbers each equal to y must be taken to make 15xy
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