# 1.8 The real numbers  (Page 6/13)

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 0.31 0.308 Convert to fractions. $\frac{31}{100}$ $\frac{308}{1000}$ We need a common denominator to compare them. $\frac{310}{1000}$ $\frac{308}{1000}$

Because 310>308, we know that $\frac{310}{1000}>\frac{308}{1000}.$ Therefore, 0.31>0.308.

Notice what we did in converting 0.31 to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000}.$ Converting $\frac{310}{1000}$ back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

$\frac{31}{100}=\frac{310}{1000}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0.31=0.310$

We say 0.31 and 0.310 are equivalent decimals    .

## Equivalent decimals

Two decimals are equivalent if they convert to equivalent fractions.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

## Order decimals.

1. Write the numbers one under the other, lining up the decimal points.
2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

Order $0.64___0.6$ using $<$ or $>.$

## Solution

$\begin{array}{cccccc}\begin{array}{c}\text{Write the numbers one under the other,}\hfill \\ \text{lining up the decimal points}.\hfill \end{array}\hfill & & & & & \begin{array}{c}\phantom{\rule{1.4em}{0ex}}0.64\hfill \\ \phantom{\rule{1.4em}{0ex}}0.6\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{Add a zero to 0.6 to make it a decimal}\hfill \\ \text{with 2 decimal places.}\hfill \end{array}\hfill & & & & & \hfill \begin{array}{c}\hfill 0.64\hfill \\ \hfill 0.60\hfill \end{array}\hfill \\ \text{Now they are both hundredths.}\hfill & & & & & \\ \\ \\ \text{64 is greater than 60.}\hfill & & & & & \hfill 64>60\hfill \\ \\ \\ \text{64 hundredths is greater than 60 hundredths.}\hfill & & & & & \hfill 0.64>0.60\hfill \\ \\ \\ & & & & & \hfill 0.64>0.6\hfill \end{array}$

Order each of the following pairs of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.42___0.4.$

>

Order each of the following pairs of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.18___0.1.$

>

Order $0.83___0.803$ using $<$ or $>.$

## Solution

$\begin{array}{cccccc}& & & & & 0.83___0.803\hfill \\ \\ \\ \begin{array}{c}\text{Write the numbers one under the other,}\hfill \\ \text{lining up the decimals.}\hfill \end{array}\hfill & & & & & \begin{array}{c}0.83\hfill \\ 0.803\hfill \end{array}\hfill \\ \\ \\ \begin{array}{c}\text{They do not have the same number of}\hfill \\ \text{digits.}\hfill \end{array}\hfill & & & & & \begin{array}{c}0.830\hfill \\ 0.803\hfill \end{array}\hfill \\ \text{Write one zero at the end of}\phantom{\rule{0.2em}{0ex}}0.83.\hfill & & & & & \\ \\ \\ \begin{array}{c}\text{Since}\phantom{\rule{0.2em}{0ex}}830>803,830\phantom{\rule{0.2em}{0ex}}\text{thousandths is}\hfill \\ \text{greater than 803 thousandths.}\hfill \end{array}\hfill & & & & & 0.830>0.803\hfill \\ \\ \\ & & & & & 0.83>0.803\hfill \end{array}$

Order the following pair of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.76___0.706.$

>

Order the following pair of numbers, using $<\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}>\text{:}\phantom{\rule{0.2em}{0ex}}0.305___0.35.$

<

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $-2$ lies to the right of $-3$ on the number line, we know that $-2>-3.$ Similarly, smaller numbers lie to the left on the number line. For example, because $-9$ lies to the left of $-6$ on the number line, we know that $-9<-6.$ See [link] .

If we zoomed in on the interval between 0 and $-1,$ as shown in [link] , we would see in the same way that $-0.2>-0.3\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-0.9<-0.6.$

Use $<$ or $>$ to order $-0.1___-0.8.$

## Solution

$\begin{array}{cccccc}& & & & & -0.1___-0.8\hfill \\ \\ \\ \begin{array}{c}\text{Write the numbers one under the other, lining up the}\hfill \\ \text{decimal points.}\hfill \end{array}\hfill & & & & & \begin{array}{c}-0.1\hfill \\ -0.8\hfill \end{array}\hfill \\ \text{They have the same number of digits.}\hfill & & & & & \\ \\ \\ \text{Since}\phantom{\rule{0.2em}{0ex}}-1>-8,-1\phantom{\rule{0.2em}{0ex}}\text{tenth is greater than}\phantom{\rule{0.2em}{0ex}}-8\phantom{\rule{0.2em}{0ex}}\text{tenths.}\hfill & & & & & -0.1>-0.8\hfill \end{array}$

Order the following pair of numbers, using<or>: $-0.3___-0.5.$

>

Order the following pair of numbers, using<or>: $-0.6___-0.7.$

>

## Key concepts

• Square Root Notation
$\sqrt{m}$ is read ‘the square root of m .’ If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge 0.$
• Order Decimals
1. Write the numbers one under the other, lining up the decimal points.
2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
3. Compare the numbers as if they were whole numbers.
4. Order the numbers using the appropriate inequality sign.

Aziza is solving this equation-2(1+x)=4x+10
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