1.15: Real Number Comparisons
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 decimal numbers. With a calculator, it is easy to express every number as a decimal number. 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 decimal numbers 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*}
\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.
Example B
For each given pair of real numbers, find another real number that is between each of the pairs.
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*}
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 \begin{align*}\frac{1}{3}\end{align*} than 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*}
The fractions do not have a common denominator. This makes it almost impossible to arrange the fractions from least to greatest. To determine the common denominator, may take some time. Let’s use the TI-83 to order these fractions.
The fractions were entered into the calculator as division problems. The decimal 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*}
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.
Now that all the numbers are in decimal form, make two lists of decimal numbers – negatives and positives. The most places after the decimal point in the given numbers is 6. The decimal numbers that you determined with your calculator need only have 6 numbers after the decimal point.
\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 inequality is a mathematical statement relating expressions by using one or more inequality symbols. The inequality symbols are \begin{align*}>,<,\ge,\le\end{align*}
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list \begin{align*}\ldots, -3, -2, -1, 0, 1, 2, 3 \ldots\end{align*}
- Irrational Numbers
- The irrational numbers are those that cannot be expressed as the ratio of two numbers. The irrational numbers include decimal numbers that are non-terminating decimals as well as non-periodic decimal numbers.
- Natural Numbers
- The natural numbers 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.
- Number Line
- A number line 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.
- Rational Numbers
- The rational numbers 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.
- Real Numbers
- The rational numbers and the irrational numbers make up the real numbers.
- Set Notation
- Set notation is a mathematical statement that shows an inequality and the set of numbers to which the variable belongs. \begin{align*}\{x|x \ge -3, x \ \varepsilon \ I\}\end{align*} is an example of set notation.
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 each of the pairs.
- 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*}
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*}
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 each of the pairs.
- \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*}
Image Attributions
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 decimal numbers. In addition, you will learn to order real numbers from least to greatest and to place these numbers on a number line. When placing numbers on a number line, you will learn helpful hints to make the process easier. Finally, you will learn to order the numbers using your TI-83 calculator.
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