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# Order of Real Numbers

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

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Practice Order of Real Numbers
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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.

### Watch This

Watch the middle part of this video, from about 3:15 to 7:30.

### Guidance

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.

#### Example A

Plot 1.25,72\begin{align*}1.25, \frac{7}{2}\end{align*}, and 26\begin{align*}2\sqrt{6}\end{align*} on a number line.

Solution: One way to compare numbers is to use a number line. To plot these numbers, convert them all to decimals. 1.25,72=3.5\begin{align*}1.25, \frac{7}{2}=3.5\end{align*}, and 264.899\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.

#### Example B

List 34,1.23,2,23,1\begin{align*}\frac{3}{4}, 1.23, \sqrt{2}, \frac{2}{3}, 1\end{align*} and 87\begin{align*}\frac{8}{7}\end{align*} in order from least to greatest.

Solution: First, write each number as a decimal.

34=0.75,1.23,21.4142,23=0.66¯,1,87=1.142857¯¯¯¯¯¯¯¯¯¯\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: 0.6667,0.75,1,1.1428,1.23,1.4142\begin{align*}0.6667, 0.75, 1, 1.1428, 1.23, 1.4142\end{align*}

Finally, exchange the decimals with the original numbers: 23,34,1,87,1.23,2\begin{align*}\frac{2}{3}, \frac{3}{4}, 1,\frac{8}{7},1.23,\sqrt{2}\end{align*}

#### Example C

Replace the blank between 53\begin{align*}-\frac{5}{3}\end{align*} ______ π2\begin{align*}-\frac{\pi}{2}\end{align*} with <, > or =.

Solution: Write both numbers in decimals. 53=1.66¯,π21.57079\begin{align*}-\frac{5}{3}=-1.6\overline{6},-\frac{\pi}{2} \approx -1.57079\end{align*}. This means that π2\begin{align*}-\frac{\pi}{2}\end{align*} is the larger number, so 53<π2\begin{align*}-\frac{5}{3}<-\frac{\pi}{2}\end{align*}.

Intro Problem Revisit 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 60÷4=15\begin{align*}60 \div 4 = 15\end{align*}. And there are 60 second in a minute, so 650 second is 650÷60=10.8333333...\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.

### Guided Practice

1. List 14,32,3,35\begin{align*}-\frac{1}{4},\frac{3}{2},-\sqrt{3},\frac{3}{5}\end{align*}, and 2 in order from greatest to least.

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

1. Write all the real numbers as decimals. 14=0.25,32=1.5,31.732,35=0.6,2\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: 2,32,35,14,3\begin{align*}2, \frac{3}{2},\frac{3}{5},-\frac{1}{4},-\sqrt{3}\end{align*}

2. 72.646\begin{align*}\sqrt{7} \approx 2.646\end{align*}. Therefore, it is larger than 2.5. Comparing the two numbers, we have 7>2.5\begin{align*}\sqrt{7} > 2.5\end{align*}.

### Explore More

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

1. 1,0.3,2\begin{align*}-1,0.3,\sqrt{2}\end{align*}
2. 14,212,3.15\begin{align*}-\frac{1}{4},-2\frac{1}{2},3.15\end{align*}
3. 1.4,56,9\begin{align*}1.4,\frac{5}{6},\sqrt{9}\end{align*}
4. 6,43,π\begin{align*}-\sqrt{6},\frac{4}{3}, \pi\end{align*}

Order the following sets of numbers from least to greatest.

1. 4,92,13,14,π\begin{align*}-4,-\frac{9}{2},-\frac{1}{3},-\frac{1}{4},-\pi\end{align*}
2. 0,12,45,16,13\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. 3.68,412,5,31112,10\begin{align*}3.68,4 \frac{1}{2},5,3 \frac{11}{12},\sqrt{10}\end{align*}
2. 2,65,114,5,3\begin{align*}-2,-\frac{6}{5},-\frac{11}{4},-\sqrt{5},-\sqrt{3}\end{align*}

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

1. 1438\begin{align*}-\frac{1}{4} \underline{\;\;\;\;\;\;\;} -\frac{3}{8}\end{align*}
2. 82.9\begin{align*}\sqrt{8}\underline{\;\;\;\;\;\;\;} 2.9\end{align*}
3. 2892.75\begin{align*}-2 \frac{8}{9} \underline{\;\;\;\;\;\;\;}-2.75\end{align*}
4. 1015812\begin{align*}\frac{10}{15} \underline{\;\;\;\;\;\;\;}\frac{8}{12}\end{align*}
5. 5052\begin{align*}-\sqrt{50} \underline{\;\;\;\;\;\;\;}-5\sqrt{2}\end{align*}
6. 1561.95\begin{align*}1 \frac{5}{6} \underline{\;\;\;\;\;\;\;}1.95\end{align*}
7. Calculator Challenge Locate the button e\begin{align*}e\end{align*} on your scientific calculator. e\begin{align*}e\end{align*} is called the natural number and will be used in the Exponential and Logarithmic Functions chapter.
1. Press the e\begin{align*}e\end{align*} button. What is e\begin{align*}e\end{align*} equivalent to?
2. What type of real number do you think e\begin{align*}e\end{align*} is?
3. Which number is larger? e\begin{align*}e\end{align*} or π\begin{align*}\pi\end{align*}?
4. Which number is larger? e\begin{align*}e\end{align*} or 7\begin{align*}\sqrt{7}\end{align*}?