# 1.8 The real numbers  (Page 6/13)

 Page 6 / 13
 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.

a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon? Mohamed Reply (a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r) muhammad Reply 4x-7y=8 2x-7y=1 what is the answer? Ramil Reply x=7/2 & y=6/7 Pbp x=7/2 & y=6/7 use Elimination Debra true bismark factoriz e usman 4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2 Peggie Ok cool answer peggie Frank thanks Ramil copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x Sandra Reply What is c+4=8 Penny Reply 2 Letha 4 Lolita 4 Rich 4 thinking C+4=8 -4 -4 C =4 thinking I need to study Letha 4+4=8 William During two years in college, a student earned$9,500. The second year, she earned $500 more than twice the amount she earned the first year. Nicole Reply 9500=500+2x Debra 9500-500=9000 9000÷2×=4500 X=4500 Debra X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500 Pbp Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles. Josh Reply Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than$500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is$7.
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The sum of two numbers is 155. Their difference is 23. Find the numbers
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