# 1.8 The real numbers  (Page 7/13)

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## Practice makes perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{36}$

6

$\sqrt{4}$

$\sqrt{64}$

8

$\sqrt{169}$

$\sqrt{9}$

3

$\sqrt{16}$

$\sqrt{100}$

10

$\sqrt{144}$

$\text{−}\sqrt{4}$

$-2$

$\text{−}\sqrt{100}$

$\text{−}\sqrt{1}$

$-1$

$\text{−}\sqrt{121}$

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

5 3.19

$\frac{5}{1}$ $\frac{319}{100}$

8 1.61

$\text{−}12\phantom{\rule{0.2em}{0ex}}$ 9.279

$\frac{-12}{1}$ $\frac{9297}{1000}$

$\text{−}16$ 4.399

In the following exercises, list the rational numbers, irrational numbers

$0.75,0.22\stackrel{\text{–}}{3},1.39174$

$0.75,0.22\stackrel{\text{–}}{3}$ $1.39174\text{…}$

$0.36,0.94729\text{…},2.52\stackrel{\text{–}}{8}$

$0.4\stackrel{\text{–}}{5},1.919293\text{…},3.59$

$0.4\stackrel{\text{–}}{5},3.59$ $1.919293\text{…}$

$0.1\stackrel{\text{–}}{3},0.42982\text{…},1.875$

In the following exercises, identify whether each number is rational or irrational.

$\sqrt{25}$ $\sqrt{30}$

rational irrational

$\sqrt{44}$ $\sqrt{49}$

$\sqrt{164}$ $\sqrt{169}$

irrational rational

$\sqrt{225}$ $\sqrt{216}$

In the following exercises, identify whether each number is a real number or not a real number.

$\text{−}\sqrt{81}$ $\sqrt{-121}$

real number not a real number

$\text{−}\sqrt{64}$ $\sqrt{-9}$

$\sqrt{-36}$ $\text{−}\sqrt{144}$

not a real number real number

$\sqrt{-49}$ $\text{−}\sqrt{144}$

In the following exercises, list the whole numbers, integers, rational numbers, irrational numbers, real numbers for each set of numbers.

$-8,0,1.95286\text{…},\frac{12}{5},\sqrt{36},9$

$0,\sqrt{36},9$ $-8,\sqrt{36},9$ $-8,0,\frac{12}{5},\sqrt{36},9$ $1.95286\text{…}$ $-8,0,1.95286\text{…},\frac{12}{5},\sqrt{36},9$

$-9,-3\frac{4}{9},\text{−}\sqrt{9},0.40\stackrel{\text{–}}{9},\frac{11}{6},7$

$\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$

none $\text{−}\sqrt{100},-7,-1$ $\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$ none $\text{−}\sqrt{100},-7,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3},-1,0.77,3\frac{1}{4}$

$-6,-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},0,0.\stackrel{\text{———}}{714285},2\frac{1}{5},\sqrt{14}$

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

$\frac{3}{4},\frac{8}{5},\frac{10}{3}$

$\frac{1}{4},\frac{9}{5},\frac{11}{3}$

$\frac{3}{10},\frac{7}{2},\frac{11}{6},4$

$\frac{7}{10},\frac{5}{2},\frac{13}{8},3$

$\frac{2}{5},-\phantom{\rule{0.2em}{0ex}}\frac{2}{5}$

$\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}$

$\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}$

$\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{2}{5},1\frac{3}{4},-1\frac{3}{4},\frac{8}{3},-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$

In the following exercises, order each of the pairs of numbers, using<or>.

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$

<

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$

$-2\frac{1}{2}___-3$

>

$-1\frac{3}{4}___-2$

$-\phantom{\rule{0.2em}{0ex}}\frac{5}{12}___-\phantom{\rule{0.2em}{0ex}}\frac{7}{12}$

>

$-\phantom{\rule{0.2em}{0ex}}\frac{9}{10}___-\phantom{\rule{0.2em}{0ex}}\frac{3}{10}$

$-3___-\phantom{\rule{0.2em}{0ex}}\frac{13}{5}$

<

$-4___-\phantom{\rule{0.2em}{0ex}}\frac{23}{6}$

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

0.8

$-0.9$

$-1.6$

3.1

In the following exercises, order each pair of numbers, using<or>.

$0.37___0.63$

<

$0.86___0.69$

$0.91___0.901$

>

$0.415___0.41$

$-0.5___-0.3$

<

$-0.1___-0.4$

$-0.62___-0.619$

<

$-7.31___-7.3$

## Everyday math

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

How many busses will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

4 busses answers may vary answers may vary

Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

How many teachers will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

## Writing exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

On a scale of $1-10,$ how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

#### Questions & Answers

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?
6.25
Ciid
6.25
Big
Wayne is hanging a string of lights 57 feet long around the three sides of his patio, which is adjacent to his house. the length of his patio, the side along the house, is 5 feet longer than twice it's width. Find the length and width of the patio.
Ciid
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
(sin=opp/adj) (tan= opp/adj) cos=hyp/adj dont quote me on it look it up
tyler
SOH = Sine is Opposite over Hypotenuse. CAH= Cosine is Adjacent over Hypotenuse. TOA = Tangent is Opposite over Adjacent.
tyler
H=57 and O=285 figure out what the adjacent?
tyler
Amara currently sells televisions for company A at a salary of $17,000 plus a$100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a$20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
what is the quantity and price of the televisions for both options?
karl
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000 17000+
Ciid
Amara has to sell 120 televisions to make 29,000 of the salary of company B. 120 * TV 100= commission 12,000+ 17,000 = of company a salary 29,000
Ciid
I'm mathematics teacher from highly recognized university.
here a question professor How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths as many sailors as soldiers? can you write out the college you went to with the name of the school you teach at and let me know the answer I've got it to be honest with you
tyler
is anyone else having issues with the links not doing anything?
Yes
Val
chapter 1 foundations 1.2 exercises variables and algebraic symbols
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold? Enter the answers in decimal form.
Joseph would like to make 12 pounds of a coffee blend at a cost of $6.25 per pound. He blends Ground Chicory at$4.40 a pound with Jamaican Blue Mountain at $8.84 per pound. How much of each type of coffee should he use? Samer 4x6.25=$25 coffee blend 4×4.40= $17.60 ground chicory 4x8.84= 35.36 blue mountain. In total they will spend for 12 pounds$77.96 they will spend in total
tyler
DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get to the park. Fabian’s speed was six miles per hour faster than DaMarcus’ speed. Find the speed of both soccer players.
i need help how to do this is confusing
what kind of math is it?
Danteii
help me to understand
huh, what is the algebra problem
Daniel
How many soldiers are there in a group of 27 sailors and soldiers if there are four fifths many sailors as soldiers?
tyler
What is the domain and range of heaviside
What is the domain and range of Heaviside and signum
Christopher
25-35
Fazal
The hypotenuse of a right triangle is 10cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle.
Tickets for a show are $70 for adults and$50 for children. For one evening performance, a total of 300 tickets were sold and the receipts totaled \$17,200. How many adult tickets and how many child tickets were sold?
A 50% antifreeze solution is to be mixed with a 90% antifreeze solution to get 200 liters of a 80% solution. How many liters of the 50% solution and how many liters of the 90% solution will be used?
June needs 45 gallons of punch for a party and has 2 different coolers to carry it in. The bigger cooler is 5 times as large as the smaller cooler. How many gallons can each cooler hold?