# 1.8 The real numbers  (Page 3/13)

 Page 3 / 13
$\text{These decimals either stop or repeat.}$

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.

find the solution to the following functions, check your solutions by substitution. f(x)=x^2-17x+72
Aziza is solving this equation-2(1+x)=4x+10
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
.473
Angelita
?
Angelita
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
p-2/3=5/6 how do I solve it with explanation pls
P=3/2
Vanarith
1/2p2-2/3p=5p/6
James
Cindy
4.5
Ruth
is y=7/5 a solution of 5y+3=10y-4
yes
James
Cindy
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have? Rhonda Reply Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 . Gabriel Reply is that a negative 4 or positive 4? Felix y = mx + b Felix if negative -4, then -4=8/9(0) + b Felix -4=b Felix if positive 4, then 4=b Felix then plug in y=8/9x - 4 or y=8/9x+4 Felix Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost$9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost$6.50 per pound to make?
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up Makaila Reply$1750
Cindy
the sum of two Numbers is 19 and their difference is 15
2, 17
Jose
interesting
saw
4,2
Cindy
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 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
hola saben como aser un valor de la expresión
NAILEA