# 1.8 The real numbers  (Page 4/13)

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Can we simplify $\sqrt{-25}?$ Is there a number whose square is $-25?$

${\left(\text{}\phantom{\rule{0.5em}{0ex}}\right)}^{2}=-25?$

None of the numbers that we have dealt with so far has a square that is $-25.$ Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to $\sqrt{-25}.$

The square root of a negative number is not a real number.

For each number given, identify whether it is a real number or not a real number: $\sqrt{-169}$ $\text{−}\sqrt{64}.$

1. There is no real number whose square is $-169.$ Therefore, $\sqrt{-169}$ is not a real number.
2. Since the negative is in front of the radical, $\text{−}\sqrt{64}$ is $-8,$ Since $-8$ is a real number, $\text{−}\sqrt{64}$ is a real number.

For each number given, identify whether it is a real number or not a real number: $\sqrt{-196}$ $\text{−}\sqrt{81}.$

not a real number real number

For each number given, identify whether it is a real number or not a real number: $\text{−}\sqrt{49}$ $\sqrt{-121}.$

real number not a real number

Given the numbers $-7,\frac{14}{5},8,\sqrt{5},5.9,\text{−}\sqrt{64},$ list the whole numbers integers rational numbers irrational numbers real numbers.

Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so $\text{−}\sqrt{64}=-8.$ So the integers are $-7,8,\text{−}\sqrt{64}.$
Since all integers are rational, then $-7,8,\text{−}\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}5.9$ are rational. So the list of rational numbers is $-7,\frac{14}{5},8,5.9,-\sqrt{64}.$
Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.
All the numbers listed are real numbers.

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: $-3,\text{−}\sqrt{2},0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}.$

$4,\sqrt{49}$ $-3,4,\sqrt{49}$ $-3,0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}$ $\text{−}\sqrt{2}$ $-3,\text{−}\sqrt{2},0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}$

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121},2.041975\text{…}$

$6,\sqrt{121}$ $\text{−}\sqrt{25},-1,6,\sqrt{121}$ $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121}$ $2.041975\text{…}$ $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121},2.041975\text{…}$

## Locate fractions on the number line

The last time we looked at the number line    , it only had positive and negative integers on it. We now want to include fraction    s and decimals on it.

Doing the Manipulative Mathematics activity “Number Line Part 3” will help you develop a better understanding of the location of fractions on the number line.

Let’s start with fractions and locate $\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{4}{5},3,\frac{7}{4},-\phantom{\rule{0.2em}{0ex}}\frac{9}{2},-5,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$ on the number line.

We’ll start with the whole numbers $3$ and $-5.$ because they are the easiest to plot. See [link] .

The proper fractions listed are $\frac{1}{5}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}.$ We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $\text{0 and 1.}$ The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts $\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}.$ We plot $\frac{1}{5}.$ See [link] .

Similarly, $-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$ is between 0 and $-1.$ After dividing the unit into 5 equal parts we plot $-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}.$ See [link] .

Finally, look at the improper fractions $\frac{7}{4},-\phantom{\rule{0.2em}{0ex}}\frac{9}{2},\frac{8}{3}.$ These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See [link] .

#### Questions & Answers

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3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
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Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have? Rhonda Reply Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 . Gabriel Reply is that a negative 4 or positive 4? Felix y = mx + b Felix if negative -4, then -4=8/9(0) + b Felix -4=b Felix if positive 4, then 4=b Felix then plug in y=8/9x - 4 or y=8/9x+4 Felix Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost$9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost$6.50 per pound to make?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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