# 1.8 The real numbers

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By the end of this section, you will be able to:
• Simplify expressions with square roots
• Identify integers, rational numbers, irrational numbers, and real numbers
• Locate fractions on the number line
• Locate decimals on the number line

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers .

## Simplify expressions with square roots

Remember that when a number n is multiplied by itself, we write ${n}^{2}$ and read it “n squared.” The result is called the square of n . For example,

$\begin{array}{cccc}{8}^{2}\hfill & & & \text{read}\phantom{\rule{0.2em}{0ex}}\text{‘}8\phantom{\rule{0.2em}{0ex}}\text{squared’}\hfill \\ 64\hfill & & & 64\phantom{\rule{0.2em}{0ex}}\text{is called the}\phantom{\rule{0.2em}{0ex}}\mathit{\text{square}}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}8.\hfill \end{array}$

Similarly, 121 is the square of 11, because ${11}^{2}$ is 121.

## Square of a number

If ${n}^{2}=m,$ then m is the square of n .

Doing the Manipulative Mathematics activity “Square Numbers” will help you develop a better understanding of perfect square numbers.

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

$\begin{array}{cccccccccc}{\left(-3\right)}^{2}=9\hfill & & & {\left(-8\right)}^{2}=64\hfill & & & {\left(-11\right)}^{2}=121\hfill & & & {\left(-15\right)}^{2}=225\hfill \end{array}$

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because ${10}^{2}=100,$ we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is $m$ is called a square root of m .

## Square root of a number

If ${n}^{2}=m,$ then n is a square root of m .

Notice ${\left(-10\right)}^{2}=100$ also, so $-10$ is also a square root of 100. Therefore, both 10 and $-10$ are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign    , $\sqrt{m},$ denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because ${0}^{2}=0,$ $\sqrt{0}=0.$ Notice that zero has only one square root.

## Square root notation

$\sqrt{m}$ is read “the square root of m

If $m={n}^{2},$ then $\sqrt{m}=n,$ for $n\ge 0.$

The square root of m , $\sqrt{m},$ is the positive number whose square is m .

Since 10 is the principal square root of 100, we write $\sqrt{100}=10.$ You may want to complete the following table to help you recognize square roots.

Simplify: $\sqrt{25}$ $\sqrt{121}.$

## Solution

$\begin{array}{cccccc}& & & & & \hfill \phantom{\rule{1em}{0ex}}\sqrt{25}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{5}^{2}=25\hfill & & & & & \hfill \phantom{\rule{1em}{0ex}}5\hfill \end{array}$

$\begin{array}{cccccc}& & & & & \hfill \sqrt{121}\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}{11}^{2}=121\hfill & & & & & \hfill 11\hfill \end{array}$

Simplify: $\sqrt{36}$ $\sqrt{169}.$

6 13

Simplify: $\sqrt{16}$ $\sqrt{196}.$

4 14

We know that every positive number has two square roots and the radical sign indicates the positive one. We write $\sqrt{100}=10.$ If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $\text{−}\sqrt{100}=-10.$ We read $\text{−}\sqrt{100}$ as “the opposite of the square root of 10.”

Simplify: $\text{−}\sqrt{9}$ $\text{−}\sqrt{144}.$

1. $\begin{array}{cccccc}& & & & & -\sqrt{9}\hfill \\ \text{The negative is in front of the radical sign.}\hfill & & & & & -3\hfill \end{array}$

2. $\begin{array}{cccccc}& & & & & -\sqrt{144}\hfill \\ \text{The negative is in front of the radical sign.}\hfill & & & & & -12\hfill \end{array}$

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
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
Washing his dad’s car alone, eight-year-old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take the Levi’s dad to wash the car by himself?
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Mckenzie
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Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes