# 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.

integer greater than 2 and less than 12
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
He charges $125 per job. His monthly expenses are$1,600. How many jobs must he work in order to make a profit of at least $2,400? Alicia Reply at least 20 Ayla what are the steps? Alicia 6.4 jobs Grahame 32 Grahame 1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32 Orlando He must work 32 jobs to make a profit POP what is algebra Azhar Reply repeated addition and subtraction of the order of operations. i love algebra I'm obsessed. Shemiah hi Krekar One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag? Leanna Reply they are 92 candies in the bag POP rectangular field solutions Navin Reply What is this? Donna t muqtaar the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is anas Reply ? Choli a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190 Usman Stella bought a dinette set on sale for$725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount? Manhwa Reply 44.19% Scott 40.22% Terence 44.2% Orlando I don't know Donna if you want the discounted price subtract$725 from $1299. then divide the answer by$1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif
Tausif
how
muqtaar
Can you explain further
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was$1,585. What is the amount of each loan?
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
think about it like you lost $19 (-19), then found$7(+7). Totally you lost just $12 (-12) Annushka I used to struggle a lot with negative numbers and math in general what I typically do is look at it in terms of money I have -$5 in my account I then take out 5 more dollars how much do I have in my account well-\$10 ... I also for a long time would draw it out on a number line to visualize it
Meg
practicing with smaller numbers to understand then working with larger numbers helps too and the song/rhyme same sign add and keep opposite signs subtract keep the sign of the bigger # then you'll be exact
Meg
hi
albert
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
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