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Can we simplify $\sqrt{\mathrm{-25}}?$ Is there a number whose square is $\mathrm{-25}?$
None of the numbers that we have dealt with so far has a square that is $\mathrm{-25}.$ Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to $\sqrt{\mathrm{-25}}.$
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: ⓐ $\sqrt{\mathrm{-169}}$ ⓑ $\text{\u2212}\sqrt{64}.$
For each number given, identify whether it is a real number or not a real number: ⓐ $\sqrt{\mathrm{-196}}$ ⓑ $\text{\u2212}\sqrt{81}.$
ⓐ not a real number ⓑ real number
For each number given, identify whether it is a real number or not a real number: ⓐ $\text{\u2212}\sqrt{49}$ ⓑ $\sqrt{\mathrm{-121}}.$
ⓐ real number ⓑ not a real number
Given the numbers $\mathrm{-7},\frac{14}{5},8,\sqrt{5},5.9,\text{\u2212}\sqrt{64},$ 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
$\mathrm{-7}$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so
$\text{\u2212}\sqrt{64}=\mathrm{-8}.$ So the integers are
$\mathrm{-7},8,\text{\u2212}\sqrt{64}.$
ⓒ Since all integers are rational, then
$\mathrm{-7},8,\text{\u2212}\sqrt{64}$ 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
$\mathrm{-7},\frac{14}{5},8,5.9,-\sqrt{64}.$
ⓓ 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: $\mathrm{-3},\text{\u2212}\sqrt{2},0.\stackrel{\text{\u2013}}{3},\frac{9}{5},4,\sqrt{49}.$
ⓐ $4,\sqrt{49}$ ⓑ $\mathrm{-3},4,\sqrt{49}$ ⓒ $\mathrm{-3},0.\stackrel{\text{\u2013}}{3},\frac{9}{5},4,\sqrt{49}$ ⓓ $\text{\u2212}\sqrt{2}$ ⓔ $\mathrm{-3},\text{\u2212}\sqrt{2},0.\stackrel{\text{\u2013}}{3},\frac{9}{5},4,\sqrt{49}$
For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: $\text{\u2212}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},\mathrm{-1},6,\sqrt{121},2.041975\text{\u2026}$
ⓐ $6,\sqrt{121}$ ⓑ $\text{\u2212}\sqrt{25},\mathrm{-1},6,\sqrt{121}$ ⓒ $\text{\u2212}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},\mathrm{-1},6,\sqrt{121}$ ⓓ $2.041975\text{\u2026}$ ⓔ $\text{\u2212}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},\mathrm{-1},6,\sqrt{121},2.041975\text{\u2026}$
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.
Let’s start with fractions and locate $\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{4}{5},3,\frac{7}{4},-\phantom{\rule{0.2em}{0ex}}\frac{9}{2},\mathrm{-5},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$ on the number line.
We’ll start with the whole numbers $3$ and $\mathrm{-5}.$ 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 $\mathrm{-1}.$ 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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