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# 1.15: Real Number Comparisons

Difficulty Level: Advanced Created by: CK-12
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Can you order the following real numbers from least to greatest?

227,1.234234,7,5,214,7,1.666,0,8.32,π2,π,5.38\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*}

### Comparing Real Numbers

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.

#### Let's draw a number line and place the numbers on them:

\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*}

\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.

#### Now, let's practice finding another real number that is between the given pair of numbers:

1. \begin{align*}-2,1\end{align*}

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

1. \begin{align*}3.5,3.6\end{align*}

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*}

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

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*}

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

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*}

#### Finally, let's 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*}

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

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

The calculator has sorted the data from least to greatest.

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*}

### Examples

#### Example 1

Earlier, you were asked to 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*}

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.

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*}

#### Example 2

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*}

\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*}

#### Example 3

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

1.  \begin{align*}-3,-5\end{align*}

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

1. \begin{align*}-3.4,-3.5\end{align*}

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*}

1. \begin{align*}\frac{1}{5},\frac{3}{10}\end{align*}

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*}

1. \begin{align*}-\frac{3}{4},-\frac{11}{6}\end{align*}

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*}

#### Example 4

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*}

\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*}

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*}

### Review

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

1. \begin{align*}\{0.5,0.45,0.65,0.33,0,2,0.75,0.28\}\end{align*}
2. \begin{align*}\{0.3,0.32,0.21,0.4,0.3,0,0.31\}\end{align*}
3. \begin{align*}\{-0.3,-0.32,-0.21,-0.4,-0.3,0,-0.31\}\end{align*}
4. \begin{align*}\{\frac{1}{2},-2,0,-\frac{1}{3},3,\frac{2}{3},-\frac{1}{2}\}\end{align*}
5. \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.

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

Use technology to sort the following numbers:

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

To see the Review answers, open this PDF file and look for section 1.15.

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### Vocabulary Language: English

TermDefinition
inequality An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
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.

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