2.13: 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*}
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*}
Practice
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. CK12 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*}b= base of the triangle and \begin{align*}h =\end{align*}h= height of the triangle. Determine the area of a triangle with base \begin{align*}= 3\end{align*}=3 feet and 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?
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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.