# 1.8 The real numbers  (Page 4/13)

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Can we simplify $\sqrt{-25}?$ Is there a number whose square is $-25?$

${\left(\text{}\phantom{\rule{0.5em}{0ex}}\right)}^{2}=-25?$

None of the numbers that we have dealt with so far has a square that is $-25.$ Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to $\sqrt{-25}.$

The square root of a negative number is not a real number.

For each number given, identify whether it is a real number or not a real number: $\sqrt{-169}$ $\text{−}\sqrt{64}.$

1. There is no real number whose square is $-169.$ Therefore, $\sqrt{-169}$ is not a real number.
2. Since the negative is in front of the radical, $\text{−}\sqrt{64}$ is $-8,$ Since $-8$ is a real number, $\text{−}\sqrt{64}$ is a real number.

For each number given, identify whether it is a real number or not a real number: $\sqrt{-196}$ $\text{−}\sqrt{81}.$

not a real number real number

For each number given, identify whether it is a real number or not a real number: $\text{−}\sqrt{49}$ $\sqrt{-121}.$

real number not a real number

Given the numbers $-7,\frac{14}{5},8,\sqrt{5},5.9,\text{−}\sqrt{64},$ list the whole numbers integers rational numbers irrational numbers real numbers.

Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and $-7$ is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so $\text{−}\sqrt{64}=-8.$ So the integers are $-7,8,\text{−}\sqrt{64}.$
Since all integers are rational, then $-7,8,\text{−}\sqrt{64}$ are rational. Rational numbers also include fractions and decimals that repeat or stop, so $\frac{14}{5}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}5.9$ are rational. So the list of rational numbers is $-7,\frac{14}{5},8,5.9,-\sqrt{64}.$
Remember that 5 is not a perfect square, so $\sqrt{5}$ is irrational.
All the numbers listed are real numbers.

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: $-3,\text{−}\sqrt{2},0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}.$

$4,\sqrt{49}$ $-3,4,\sqrt{49}$ $-3,0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}$ $\text{−}\sqrt{2}$ $-3,\text{−}\sqrt{2},0.\stackrel{\text{–}}{3},\frac{9}{5},4,\sqrt{49}$

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121},2.041975\text{…}$

$6,\sqrt{121}$ $\text{−}\sqrt{25},-1,6,\sqrt{121}$ $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121}$ $2.041975\text{…}$ $\text{−}\sqrt{25},-\phantom{\rule{0.2em}{0ex}}\frac{3}{8},-1,6,\sqrt{121},2.041975\text{…}$

## Locate fractions on the number line

The last time we looked at the number line    , it only had positive and negative integers on it. We now want to include fraction    s and decimals on it.

Doing the Manipulative Mathematics activity “Number Line Part 3” will help you develop a better understanding of the location of fractions on the number line.

Let’s start with fractions and locate $\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{4}{5},3,\frac{7}{4},-\phantom{\rule{0.2em}{0ex}}\frac{9}{2},-5,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$ on the number line.

We’ll start with the whole numbers $3$ and $-5.$ because they are the easiest to plot. See [link] .

The proper fractions listed are $\frac{1}{5}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}.$ We know the proper fraction $\frac{1}{5}$ has value less than one and so would be located between $\text{0 and 1.}$ The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts $\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}.$ We plot $\frac{1}{5}.$ See [link] .

Similarly, $-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$ is between 0 and $-1.$ After dividing the unit into 5 equal parts we plot $-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}.$ See [link] .

Finally, look at the improper fractions $\frac{7}{4},-\phantom{\rule{0.2em}{0ex}}\frac{9}{2},\frac{8}{3}.$ These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See [link] .

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be? 74.25 × .16 then get the total and that will be your tip David what is the shorter way to do it Cesar Reply Priam has dimes and pennies in a cup holder in his car. The total value of the coins is$4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne