# 2.13: Order of Real Numbers

**Basic**Created by: CK-12

**Practice**Order of Real Numbers

Suppose that you and three friends were playing a game where you each drew a number from a hat and the person with the highest number won. Let's say that you drew the number \begin{align*} \frac{3}{2}\end{align*}

### Guidance

**Classifying Real Numbers**

#### Example A

*Using the chart above, categorize the following numbers:*

a) 0

b) –1

c) \begin{align*}\frac{\pi}{3}\end{align*}

d) \begin{align*}\frac{\sqrt{36}}{9}\end{align*}

**Solutions:**

a) Zero is a whole number, an integer, a rational number, and a real number.

b) –1 is an integer, a rational number, and a real number.

c) \begin{align*}\frac{\pi}{3}\end{align*}

d) \begin{align*}\frac{\sqrt{36}}{9} = \frac{6}{9} = \frac{2}{3}\end{align*}

**Graphing and Ordering Real Numbers**

Every real number can be positioned between two integers. Many times you will need to organize real numbers to determine the least value, greatest value, or both. This is usually done on a number line.

#### Example B

*Plot the following rational numbers on a number line.*

a) \begin{align*}\frac{2}{3}\end{align*}

b) \begin{align*}-\frac{3}{7}\end{align*}

c) \begin{align*}\frac{57}{16}\end{align*}

**Solutions:**

a) \begin{align*}\frac{2}{3} = 0.\overline{6}\end{align*}

b) \begin{align*}-\frac{3}{7}\end{align*}

c) \begin{align*}\frac{57}{16} = 3.5625\end{align*}

#### Example C

Compare \begin{align*} \frac{\pi}{15}\end{align*}

**Solution:**

First we simplify in order to better compare:

\begin{align*}\frac{\sqrt{3}}{\sqrt{75}}=\frac{\sqrt{3}}{5\sqrt{3}}=\frac{1}{5}.\end{align*}

Now we rewrite \begin{align*} \frac{\pi}{15}\end{align*}

\begin{align*} \frac{\pi}{15}=\frac{\pi}{3\times 5}=\frac{\pi}{3}\times \frac{1}{5}.\end{align*}

Since \begin{align*}\pi >3\end{align*}

\begin{align*}\frac{\pi}{3}>1\end{align*}

so

\begin{align*}\frac{\pi}{3}\times \frac{1}{5}> \frac{1}{5}.\end{align*}

Therefore, \begin{align*}\frac{\pi}{15}> \frac{\sqrt{3}}{\sqrt{75}}.\end{align*}

### Video Review

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### Guided Practice

For the numbers: \begin{align*} \frac{\sqrt{12}}{2}, 1.5\cdot \sqrt{3}, \frac{3}{2}, \frac{2\sqrt{5}}{\sqrt{20}}\end{align*}

1. Classify each number.

2. Order the four numbers.

**Solutions:**

1. We need to simplify the numbers in order to classify them:

\begin{align*} \frac{\sqrt{12}}{2}=\frac{\sqrt{4\times 3}}{2}=\frac{2\sqrt{3}}{2}=\sqrt{3}.\end{align*}

\begin{align*}1.5\cdot \sqrt{3}. \end{align*}

\begin{align*} \frac{3}{2}. \end{align*}

\begin{align*} \frac{2\sqrt{5}}{\sqrt{20}}=\frac{2\sqrt{5}}{\sqrt{4\times 5}}=\frac{2\sqrt{5}}{2\sqrt{5}}=1.\end{align*}

2. The four numbers are ordered as follows: \begin{align*}1<\frac{3}{2}<\sqrt{3} <1.5\cdot \sqrt{3}.\end{align*}

\begin{align*}1<\frac{3}{2}\end{align*}

\begin{align*}\frac{3}{2}<\sqrt{3}\end{align*}

\begin{align*}\sqrt{3} < 1.5\cdot \sqrt{3}\end{align*}

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Square Roots and Real Numbers (10:18)

Classify the following numbers. Include all the categories that apply to the number.

- \begin{align*}\sqrt{0.25}\end{align*}
0.25−−−−√ - \begin{align*}\sqrt{1.35}\end{align*}
1.35−−−−√ - \begin{align*}\sqrt{20}\end{align*}
20−−√ - \begin{align*}\sqrt{25}\end{align*}
25−−√ - \begin{align*}\sqrt{100}\end{align*}
100−−−√ - Place the following numbers in numerical order from lowest to highest. \begin{align*}\frac{\sqrt{6}}{2} && \frac{61}{50} && \sqrt{1.5} && \frac{16}{13}\end{align*}
6√261501.5−−−√1613 - Find the value of each marked point.

**Mixed Review**

- Simplify \begin{align*}\frac{9}{4}\div 6\end{align*}
94÷6 . - The area of a triangle is given by the formula \begin{align*}A= \frac{b(h)}{2}\end{align*}
A=b(h)2 ,*where \begin{align*}b=\end{align*}*. Determine the area of a triangle withb= base of the triangle and \begin{align*}h =\end{align*}h= height of the triangle*base \begin{align*}= 3\end{align*}*and=3 feet*height \begin{align*}= 7\end{align*}*.=7 feet - Reduce the fraction \begin{align*}\frac{144}{6}\end{align*}
1446 . - Construct a table for the following situation:
*Tracey jumps 60 times per minute*. Let the minutes be \begin{align*}\left \{0,1,2,3,4,5,6\right \}\end{align*}{0,1,2,3,4,5,6} . What is the range of this function?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.13.

### Notes/Highlights Having trouble? Report an issue.

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### Image Attributions

Here you'll learn how to decide if a number is a whole number, an integer, a rational or irrational number, or a real number. You'll also learn how to put numbers in order and how to graph them on a number line.

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