# Order of Real Numbers

## Subsets of real number system. Simplify numbers before classifying

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Ordering Real Numbers

Mariah can spin a basketball on her finger for a quarter of an hour, Chita can spin one for 10 minutes, and Dakara can spin one for 650 seconds. Who can spin the basketball on her finger the longest?

We need to put all these different ways of measuring time in the same unit.

### Putting Real Numbers in Order

Real numbers can be listed in order even if they are different types of real numbers. The easiest way to do this is to convert all the real numbers into decimals.

Let's do the following problems.

1. Plot \begin{align*}1.25, \frac{7}{2}\end{align*}, and \begin{align*}2\sqrt{6}\end{align*} on a number line.

One way to compare numbers is to use a number line. To plot these numbers, convert them all to decimals. \begin{align*}1.25, \frac{7}{2}=3.5\end{align*}, and \begin{align*}2\sqrt{6}\approx 4.899\end{align*} (The symbol \begin{align*}\approx\end{align*} means approximately.) Draw your number line and plot the points. Recall that 0 is called the origin.

Depending on your scale, you can have hash marks at half-values or only even values. The placement of each number on the number line is an approximate representation of each number.

1. List \begin{align*}\frac{3}{4}, 1.23, \sqrt{2}, \frac{2}{3}, 1\end{align*} and \begin{align*}\frac{8}{7}\end{align*} in order from least to greatest.

First, write each number as a decimal.

\begin{align*}\frac{3}{4}=0.75, 1.23, \sqrt{2} \approx 1.4142, \frac{2}{3}=0.6\overline{6},1,\frac{8}{7}=1.\overline{142857}\end{align*}. Now, write the decimals, in order, starting with the smallest and ending with the largest: \begin{align*}0.6667, 0.75, 1, 1.1428, 1.23, 1.4142\end{align*}

Finally, exchange the decimals with the original numbers: \begin{align*}\frac{2}{3}, \frac{3}{4}, 1,\frac{8}{7},1.23,\sqrt{2}\end{align*}

1. Replace the blank between \begin{align*}-\frac{5}{3}\end{align*} ______ \begin{align*}-\frac{\pi}{2}\end{align*} with <, > or =.

Write both numbers in decimals. \begin{align*}-\frac{5}{3}=-1.6\overline{6},-\frac{\pi}{2} \approx -1.57079\end{align*}. This means that \begin{align*}-\frac{\pi}{2}\end{align*} is the larger number, so \begin{align*}-\frac{5}{3}<-\frac{\pi}{2}\end{align*}.

### Examples

#### Example 1

Earlier, you were asked who can spin the basketball on her finger the longest.

To compare these different measurements of time, let's put them all in minutes. A quarter of an hour is a quarter of 60 minutes or \begin{align*}60 \div 4 = 15\end{align*}. And there are 60 second in a minute, so 650 seconds is \begin{align*}650 \div 60 = 10.8333333...\end{align*} minutes. From this, we know that a quarter of an hour is the most and therefore Mariah can spin the basketball on her finger the longest.

#### Example 2

List \begin{align*}-\frac{1}{4},\frac{3}{2},-\sqrt{3},\frac{3}{5}\end{align*}, and 2 in order from greatest to least.

Write all the real numbers as decimals. \begin{align*}-\frac{1}{4}=-0.25,\frac{3}{2}=1.5,-\sqrt{3} \approx -1.732, \frac{3}{5}=0.6,2\end{align*} In order, the numbers are: \begin{align*}2, \frac{3}{2},\frac{3}{5},-\frac{1}{4},-\sqrt{3}\end{align*}

#### Example 3

Compare \begin{align*}\sqrt{7}\end{align*} and 2.5 by using <, >, or =.

\begin{align*}\sqrt{7} \approx 2.646\end{align*}. Therefore, it is larger than 2.5. Comparing the two numbers, we have \begin{align*}\sqrt{7} > 2.5\end{align*}.

### Review

Plot the following numbers on a number line. Use an appropriate scale.

1. \begin{align*}-1,0.3,\sqrt{2}\end{align*}
2. \begin{align*}-\frac{1}{4},-2\frac{1}{2},3.15\end{align*}
3. \begin{align*}1.4,\frac{5}{6},\sqrt{9}\end{align*}
4. \begin{align*}-\sqrt{6},\frac{4}{3}, \pi\end{align*}

Order the following sets of numbers from least to greatest.

1. \begin{align*}-4,-\frac{9}{2},-\frac{1}{3},-\frac{1}{4},-\pi\end{align*}
2. \begin{align*}0,-\frac{1}{2},\frac{4}{5},\frac{1}{6},-\sqrt{\frac{1}{3}}\end{align*}

Order the following sets of numbers from greatest to least.

1. \begin{align*}3.68,4 \frac{1}{2},5,3 \frac{11}{12},\sqrt{10}\end{align*}
2. \begin{align*}-2,-\frac{6}{5},-\frac{11}{4},-\sqrt{5},-\sqrt{3}\end{align*}

Compare each pair of numbers using <, >, and =.

1. \begin{align*}-\frac{1}{4} \underline{\;\;\;\;\;\;\;} -\frac{3}{8}\end{align*}
2. \begin{align*}\sqrt{8}\underline{\;\;\;\;\;\;\;} 2.9\end{align*}
3. \begin{align*}-2 \frac{8}{9} \underline{\;\;\;\;\;\;\;}-2.75\end{align*}
4. \begin{align*}\frac{10}{15} \underline{\;\;\;\;\;\;\;}\frac{8}{12}\end{align*}
5. \begin{align*}-\sqrt{50} \underline{\;\;\;\;\;\;\;}-5\sqrt{2}\end{align*}
6. \begin{align*}1 \frac{5}{6} \underline{\;\;\;\;\;\;\;}1.95\end{align*}
7. Calculator Challenge Locate the button \begin{align*}e\end{align*} on your scientific calculator. \begin{align*}e\end{align*} is called the natural numberand will be used in the Exponential and Logarithmic Functions chapter.
1. Press the \begin{align*}e\end{align*} button. What is \begin{align*}e\end{align*} equivalent to?
2. What type of real number do you think \begin{align*}e\end{align*} is?
3. Which number is larger? \begin{align*}e\end{align*} or \begin{align*}\pi\end{align*}?
4. Which number is larger? \begin{align*}e\end{align*} or \begin{align*}\sqrt{7}\end{align*}

To see the Review answers, open this PDF file and look for section 1.2.

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