# 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?

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?

In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary? Iris Reply let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is$500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for$3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound? Valeria Reply Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be$10 per square foot?
I need help with maths can someone help me plz.. is there a wats app group?
WY need
Fernando
How did you get $750? Laura Reply if y= 2x+sinx what is dy÷dx formon25 Reply does it teach you how to do algebra if you don't know how Kate Reply Liam borrowed a total of$35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total$35,000? (Round your answer to the nearest tenth of a percent.)
exact definition of length by bilbao
the definition of length
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce