1.8 The real numbers  (Page 4/13)

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.

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.

Let’s start with fractions and locate $\frac{1}{5},-\frac{4}{5},3,\frac{7}{4},-\frac{9}{2},\mathrm{-5},\text{and}\frac{8}{3}$ 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 $\frac{1}{5}\text{and}-\frac{4}{5}.$ 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, $-\frac{4}{5}$ is between 0 and $\mathrm{-1}.$ After dividing the unit into 5 equal parts we plot $-\frac{4}{5}.$ See [link] .

Finally, look at the improper fractions $\frac{7}{4},-\frac{9}{2},\frac{8}{3}.$ 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] .