# 1.8 The real numbers  (Page 3/13)

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$\text{These decimals either stop or repeat.}$

What do these examples tell us?

Every rational number can be written both as a ratio of integers , $\left(\frac{p}{q},$ where p and q are integers and $q\ne 0\right),$ and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Fractions Integers
Number $\frac{4}{5}$ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$ $\frac{13}{4}$ $-\phantom{\rule{0.2em}{0ex}}\frac{20}{3}$ $-2$ $-1$ $0$ $1$ $2$ $3$
Ratio of Integers $\frac{4}{5}$ $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}$ $\frac{13}{4}$ $-\phantom{\rule{0.2em}{0ex}}\frac{20}{3}$ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{1}$ $-\phantom{\rule{0.2em}{0ex}}\frac{1}{1}$ $\frac{0}{1}$ $\frac{1}{1}$ $\frac{2}{1}$ $\frac{3}{1}$
Decimal Form $0.8$ $-0.875$ $3.25$ $-6.\stackrel{\text{–}}{6}$ $-2.0$ $-1.0$ $0.0$ $1.0$ $2.0$ $3.0$

## Rational number

A rational number is a number of the form $\frac{p}{q},$ where p and q are integers and $q\ne 0.$

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number $\pi$ (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

$\pi =3.141592654...$

We can even create a decimal pattern that does not stop or repeat, such as

$2.01001000100001\dots$

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

## Irrational number

An irrational number    is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

## Rational or irrational?

If the decimal form of a number

• repeats or stops , the number is rational .
• does not repeat and does not stop , the number is irrational .

Given the numbers $0.58\stackrel{\text{–}}{3},0.47,3.605551275...$ list the rational numbers irrational numbers.

## Solution

$\begin{array}{cccccc}\text{Look for decimals that repeat or stop.}\hfill & & & & & \text{The}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{repeats in}\phantom{\rule{0.2em}{0ex}}0.58\stackrel{\text{–}}{3}.\hfill \\ & & & & & \text{The decimal}\phantom{\rule{0.2em}{0ex}}0.47\phantom{\rule{0.2em}{0ex}}\text{stops after the}\phantom{\rule{0.2em}{0ex}}7.\hfill \\ & & & & & \text{So}\phantom{\rule{0.2em}{0ex}}0.58\stackrel{\text{–}}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0.47\phantom{\rule{0.2em}{0ex}}\text{are rational.}\hfill \end{array}$

$\begin{array}{cccccc}\text{Look for decimals that neither stop nor repeat.}\hfill & & & & & 3.605551275\text{…}\phantom{\rule{0.2em}{0ex}}\text{has no repeating block of}\hfill \\ & & & & & \text{digits and it does not stop.}\hfill \\ & & & & & \text{So}\phantom{\rule{0.2em}{0ex}}3.605551275\text{…}\phantom{\rule{0.2em}{0ex}}\text{is irrational.}\hfill \end{array}$

For the given numbers list the rational numbers irrational numbers: $0.29,0.81\stackrel{\text{–}}{6},2.515115111\text{…}.$

$0.29,0.81\stackrel{\text{–}}{6}$ $2.515115111\text{…}$

For the given numbers list the rational numbers irrational numbers: $2.6\stackrel{\text{–}}{3},0.125,0.418302\text{…}$

$2.6\stackrel{\text{–}}{3},0.125$ $0.418302\text{…}$

For each number given, identify whether it is rational or irrational: $\sqrt{36}$ $\sqrt{44}.$

1. Recognize that 36 is a perfect square, since ${6}^{2}=36.$ So $\sqrt{36}=6,$ therefore $\sqrt{36}$ is rational.
2. Remember that ${6}^{2}=36$ and ${7}^{2}=49,$ so 44 is not a perfect square. Therefore, the decimal form of $\sqrt{44}$ will never repeat and never stop, so $\sqrt{44}$ is irrational.

For each number given, identify whether it is rational or irrational: $\sqrt{81}$ $\sqrt{17}.$

rational irrational

For each number given, identify whether it is rational or irrational: $\sqrt{116}$ $\sqrt{121}.$

irrational rational

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number     s .

## Real number

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. [link] illustrates how the number sets we’ve discussed in this section fit together.

DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players.
?
Ann
DaMarcus: 16 mi/hr Fabian: 22 mi/hr
Sherman
Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution?
15 and 5
32 is 40% , & 8 is 10 % , & any 4 letters is 5%.
Karen
It felt that something is missing on the question like: 40% of what solution? 10% of what solution?
Jhea
its confusing
Sparcast
3% & 2% to complete the 25%
Sparcast
because she already has 20 liters.
Sparcast
ok I was a little confused I agree 15% & 5%
Sparcast
8,2
Karen
Jim and Debbie earned $7200. Debbie earned$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike
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?
1,75hrs
Mike
I'm going to guess. Divide Levi's time by 2. Then divide 1 hour by 2. 1.25 + 0.5 = 1.3?
John
Oops I mean 1.75
John
I'm guessing this because since I have divide 1 hour by 2, I have to do the same for the 2.5 hours it takes Levi by himself.
John
1,75 hrs is correct Mike
Emund
How did you come up with the answer?
John
Drew burned 1,800 calories Friday playing 1 hour of basketball and canoeing for 2 hours. On Saturday, he spent 2 hours playing basketball and 3 hours canoeing and burned 3,200 calories. How many calories did he burn per hour when playing basketball?
Brandon has a cup of quarters and dimes with a total value of $3.80. The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have? Kendra Reply Tickets to a Broadway show cost$35 for adults and $15 for children. The total receipts for 1650 tickets at one performance were$47,150. How many adult and how many child tickets were sold?
825
Carol
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year? How do I do this
how to square
easiest way to find the square root of a large number?
Jackie
the accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery set up a linear system whose solution provides the unknown flow rates (b) solve the system for the unknown flow rates (c) find the flow rates and directions of flow if x4=50and x6=0
What is observation
I'm confused by the question. Can you describe or explain the math question it pertains to?
Melissa
there is no math to it because all you use is your vision or gaze to the sorrounding areas
Cesarp
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him$10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
5 times max
Anton
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 1/3 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
35 min
Debra