# 1.15: Real Number Comparisons

**Advanced**Created by: CK-12

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**Practice**Sets and Symbols

Can you order the following real numbers from least to greatest?

\begin{align*}\frac{22}{7},1.234 234 \ldots, - \sqrt{7}, -5, -\frac{21}{4}, 7,-1.666,0,8.32,\frac{\pi}{2},-\pi,-5.38\end{align*}

### Watch This

Khan Academy Points on a Number Line

### Guidance

The simplest way to order numbers is to express them all in the same form – all fractions or all decimals. With a calculator, it is easy to express every number in its decimal form. Watch your signs – don't drop any of the negative signs.

When plotting numbers on a number line, keep in mind that it is impossible to place decimals in the exact location on the number line. Place them as close as you can to the appropriate spot on the line.

#### Example A

Draw a number line and place these numbers on the line.

\begin{align*}{\color{red}\sqrt{\frac{2}{5}}}, {\color{blue}0.6467},{\color{red}-\frac{3}{5}},{\color{red}\frac{1}{8}},{\color{green}0},{\color{red}\sqrt{0.5}},{\color{blue}-2.34},{\color{red}\pi},{\color{red}\frac{2 \pi}{3},{\color{green}-1}},{\color{green}2}\end{align*}

**Solution:** \begin{align*}\sqrt{\frac{2}{5}}=0.6324... \quad -\frac{3}{5}=-0.6 \quad \frac{1}{8}=0.125 \quad \sqrt{0.5}=0.7071... \quad -\pi=-3.1416... \quad \frac{2 \pi}{3}=2.0944...\end{align*}

Start by placing the \begin{align*}{\color{green}\mathbf{integers}}\end{align*} on the line first. Next place the \begin{align*}{\color{blue}\mathbf{decimal \ numbers}}\end{align*} on the line.

Use your calculator to convert \begin{align*}{\color{red}\mathbf{the \ remaining \ numbers}}\end{align*} to decimal numbers. Place these on the line last.

**License**: CC BY-NC 3.0

#### Example B

For each given pair of real numbers, find another real number that is between the pair of numbers.

i) \begin{align*}-2,1\end{align*}

ii) \begin{align*}3.5,3.6\end{align*}

iii) \begin{align*}\frac{1}{2},\frac{1}{3}\end{align*}

iv) \begin{align*}-\frac{1}{3}, -\frac{1}{4}\end{align*}

**Solution:** There are multiple possible answers. Here are possible answers.

i) The number must be greater than –2 and less than 1. \begin{align*}-2, {\color{blue}0},1\end{align*}

ii) The number must be greater than 3.5 and less than 3.6. \begin{align*}3.5, {\color{blue}3.54},3.6\end{align*}

iii) The number must be greater than \begin{align*}\frac{1}{3}\end{align*} and less than \begin{align*}\frac{1}{2}\end{align*}. Write each fraction with a common denominator. \begin{align*}\frac{1}{2}=\frac{3}{6},\frac{1}{3}=\frac{2}{6}\end{align*}. If you look at \begin{align*}\frac{2}{6}\end{align*} and \begin{align*}\frac{3}{6}\end{align*}, there is no fraction, with a denominator of 6, between these values. Write the fractions with a larger common denominator. \begin{align*}\frac{1}{2}=\frac{6}{12}, \frac{1}{3}=\frac{4}{12}\end{align*}. If you look at \begin{align*}\frac{4}{12}\end{align*} and \begin{align*}\frac{6}{12}\end{align*}, the fraction \begin{align*}\frac{5}{12}\end{align*} is between them. \begin{align*}\frac{1}{3},{\color{blue}\frac{5}{12}},\frac{1}{2}\end{align*}

iv) The number must be greater than \begin{align*}-\frac{1}{3}\end{align*} and less than \begin{align*}-\frac{1}{4}\end{align*}. Write each fraction with a common denominator. \begin{align*}-\frac{1}{3}=-\frac{4}{12},-\frac{1}{4}=-\frac{3}{12}\end{align*}. If you look at \begin{align*}-\frac{3}{12}\end{align*} and \begin{align*}-\frac{4}{12}\end{align*}, there is no fraction, with a denominator of 12, between these values. Write the fractions with a larger common denominator. \begin{align*}-\frac{1}{3}=-\frac{8}{24},-\frac{1}{4}=-\frac{6}{24}\end{align*}. If you look at \begin{align*}-\frac{6}{24}\end{align*} and \begin{align*}-\frac{8}{24}\end{align*}, the fraction \begin{align*}-\frac{7}{24}\end{align*} is between them. \begin{align*}-\frac{8}{24}, {\color{blue}-\frac{7}{24}}, -\frac{6}{24}\end{align*}

#### Example C

Order the following fractions from least to greatest.

\begin{align*}\frac{2}{11},\frac{7}{9},\frac{8}{7},\frac{1}{11},\frac{5}{6}\end{align*}

**Solution:** The fractions do not have a common denominator. Let’s use the TI-83 to order these fractions.

**License**: CC BY-NC 3.0

The fractions were entered into the calculator as division problems. The decimal forms of the numbers were entered into List 1.

**License**: CC BY-NC 3.0

The calculator has sorted the data from least to greatest.

**License**: CC BY-NC 3.0

The data is sorted. The decimal numbers and the corresponding fractions can now be matched from the screen where they were first entered.

\begin{align*}\frac{1}{11},\frac{2}{11},\frac{7}{9},\frac{5}{6},\frac{8}{7}\end{align*}

#### Concept Problem Revisited

\begin{align*}\frac{22}{7},1.234 234 \ldots, - \sqrt{7}, -5, -\frac{21}{4}, 7,-1.666,0,8.32,\frac{\pi}{2},-\pi,-5.38\end{align*}

As you examine the above numbers, you can see that there are natural numbers, whole numbers, integers, rational numbers and irrational numbers. These numbers, as they are presented here, would be very difficult to order from least to greatest.

**License**: CC BY-NC 3.0

Now that all the numbers are in decimal form, order them from least to greatest.

\begin{align*}-5.38, -\frac{21}{4}, -5, -\pi, -\sqrt{7}, -1.666, 0, 1.234234, \frac{\pi}{2},\frac{22}{7}, 7, 8.32\end{align*}

### Vocabulary

- Inequality
- An
is a mathematical statement relating expressions by using one or more inequality symbols. The inequality symbols are \begin{align*}>,<,\ge,\le\end{align*}.*inequality*

- Integer
- All natural numbers, their opposites, and zero are
. A number in the list \begin{align*}\ldots, -3, -2, -1, 0, 1, 2, 3 \ldots\end{align*}.*integers*

- Irrational Numbers
- The
are those that cannot be expressed as the ratio of two numbers. The*irrational numbers*include decimal numbers that are both non-terminating decimals as well as non-periodic decimal numbers.*irrational numbers*

- Natural Numbers
- The
are the counting numbers and consist of all positive, whole numbers. The*natural numbers*are numbers in the list \begin{align*}1, 2, 3\ldots\end{align*} and are often referred to as positive integers.*natural numbers*

- Number Line
- A
is a line that matches a set of points and a set of numbers one to one. It is often used in mathematics to show mathematical computations.*number line*

- Rational Numbers
- The
are numbers that can be written as the ratio of two numbers \begin{align*}\frac{a}{b}\end{align*} and \begin{align*}b \ne 0\end{align*}. The*rational numbers*include all terminating decimals as well as periodic decimal numbers.*rational numbers*

- Real Numbers
- The rational numbers and the irrational numbers make up the
*real numbers.*

### Guided Practice

1. Arrange the following numbers in order from least to greatest and place them on a number line.

\begin{align*}-3.78, -\frac{11}{4},-4, \frac{\pi}{2}, -\sqrt{6},-1.888,0,0.2424,\pi,\frac{21}{15},2,1.75\end{align*}

2. For each given pair of real numbers, find another real number that is between the pair of numbers.

- i) \begin{align*}-3,-5\end{align*}
- ii) \begin{align*}-3.4,-3.5\end{align*}
- iii) \begin{align*}\frac{1}{5},\frac{3}{10}\end{align*}
- iv) \begin{align*}-\frac{3}{4},-\frac{11}{6}\end{align*}

3. Use technology to sort the following numbers:

\begin{align*}\sqrt{\frac{3}{5}},\frac{15}{38},-\frac{7}{12},\frac{1}{4},0,\sqrt{8},-\frac{13}{21},-\pi,\frac{3 \pi}{5},-6,3\end{align*}

**Answers:**

1. \begin{align*}-3.78, -\frac{11}{4},-4, \frac{\pi}{2}, -\sqrt{6},-1.888,0,0.2424,\pi,\frac{21}{15},2,1.75\end{align*}

**License**: CC BY-NC 3.0

2. i) The number must be greater than –5 and less than –3. \begin{align*}-5,{\color{blue}-4},-3\end{align*}

- ii) The number must be greater than –3.5 and less than –3.4. \begin{align*}-3.5,{\color{blue}-3.45},-3.4\end{align*}
- iii) The number must be greater than \begin{align*}\frac{1}{5}\end{align*} and less than \begin{align*}\frac{3}{10}\end{align*}. Write each fraction with a common denominator. \begin{align*}\frac{1}{5}=\frac{2}{10}\end{align*}. If you look at \begin{align*}\frac{2}{10}\end{align*} and \begin{align*}\frac{3}{10}\end{align*}, there is no fraction, with a denominator of 10, between these values. Write the fractions with a larger common denominator. \begin{align*}\frac{1}{5}=\frac{4}{20},\frac{3}{10}=\frac{6}{20}\end{align*}. If you look at \begin{align*}\frac{4}{20}\end{align*} and \begin{align*}\frac{6}{20}\end{align*}, the fraction \begin{align*}\frac{5}{20}=\frac{1}{4}\end{align*} is between them. \begin{align*}\frac{1}{5}, \frac{{\color{blue}1}}{{\color{blue}4}}, \frac{3}{10}\end{align*}
- iv) The number must be greater than \begin{align*}-\frac{3}{4}\end{align*}and less than \begin{align*}-\frac{11}{16}\end{align*}. Write each fraction with a common denominator. \begin{align*}-\frac{3}{4}=-\frac{12}{16}\end{align*}. If you look at \begin{align*}-\frac{12}{16}\end{align*} and \begin{align*}-\frac{11}{16}\end{align*}, there is no fraction, with a denominator of 16 between these values. Write the fractions with a larger common denominator. \begin{align*}-\frac{3}{4}=-\frac{24}{32},-\frac{11}{16}=-\frac{22}{32}\end{align*}. If you look at \begin{align*}-\frac{24}{32}\end{align*} and \begin{align*}-\frac{22}{32}\end{align*}, the fraction \begin{align*}-\frac{23}{32}\end{align*} is between them. \begin{align*}-\frac{3}{4},-{\color{blue}\frac{23}{32}},-\frac{11}{16}\end{align*}

3. \begin{align*}\sqrt{\frac{3}{5}},\frac{15}{38},-\frac{7}{12},\frac{1}{4},0,\sqrt{8},-\frac{13}{21},-\pi,\frac{3 \pi}{5},-6,3\end{align*}

**License**: CC BY-NC 3.0

*The numbers have been sorted. The numbers from least to greatest are:*

\begin{align*}-6,-\pi,-\frac{13}{21},-\frac{7}{12},0,\frac{1}{4},\frac{15}{38},\sqrt{\frac{3}{5}},\frac{3 \pi}{5},\sqrt{8},3\end{align*}

### Practice

Arrange the following numbers in order from least to greatest and place them on a number line.

- \begin{align*}\{0.5,0.45,0.65,0.33,0,2,0.75,0.28\}\end{align*}
- \begin{align*}\{0.3,0.32,0.21,0.4,0.3,0,0.31\}\end{align*}
- \begin{align*}\{-0.3,-0.32,-0.21,-0.4,-0.3,0,-0.31\}\end{align*}
- \begin{align*}\{\frac{1}{2},-2,0,-\frac{1}{3},3,\frac{2}{3},-\frac{1}{2}\}\end{align*}
- \begin{align*}\{0.3,-\sqrt{2},1,-0.25,0,1.8,-\frac{\pi}{3}\}\end{align*}

For each given pair of real numbers, find another real number that is between the pair of numbers.

- \begin{align*}8,10\end{align*}
- \begin{align*}-12,-13\end{align*}
- \begin{align*}-12.01,-12.02\end{align*}
- \begin{align*}-7.6,-7.5\end{align*}
- \begin{align*}\frac{1}{7},\frac{4}{21}\end{align*}
- \begin{align*}\frac{2}{5},\frac{7}{9}\end{align*}
- \begin{align*}-\frac{2}{9},-\frac{3}{18}\end{align*}
- \begin{align*}-\frac{3}{5},-\frac{1}{2}\end{align*}

Use technology to sort the following numbers:

- \begin{align*}\{-2,\frac{2}{3},0,\frac{3}{8},-\frac{7}{5},\frac{1}{2},4,-3.6\}\end{align*}
- \begin{align*}\{\sqrt{10},-1,\frac{7}{12},3,-\frac{5}{4},-\sqrt{7},0,-\frac{2 \pi}{3},-\frac{3}{5}\}\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.15.

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.Natural Numbers

The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are the numbers in the list 1, 2, 3... and are often referred to as positive integers.number line

A number line is a line on which numbers are marked at intervals. Number lines are often used in mathematics to show mathematical computations.rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.### Image Attributions

**[1]****^**License: CC BY-NC 3.0**[2]****^**License: CC BY-NC 3.0**[3]****^**License: CC BY-NC 3.0**[4]****^**License: CC BY-NC 3.0**[5]****^**License: CC BY-NC 3.0**[6]****^**License: CC BY-NC 3.0**[7]****^**License: CC BY-NC 3.0

## Description

## Learning Objectives

Here you will revisit the number sets that make up the real number system. You will also apply the skills you have learned for changing fractions to decimals. In addition, you will learn to order real numbers from least to greatest and to place these numbers on a number line.

## Related Materials

## Difficulty Level:

Advanced## Tags:

## Subjects:

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## Date Created:

Apr 30, 2013## Last Modified:

Feb 26, 2015## Vocabulary

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