# 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.

He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply rectangular field solutions Navin Reply What is this? Donna the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
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
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed