# 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] .

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Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
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?