# 1.8 The real numbers  (Page 2/13)

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Simplify: $\text{−}\sqrt{4}$ $\text{−}\sqrt{225}.$

$-2$ $-15$

Simplify: $\text{−}\sqrt{81}$ $\text{−}\sqrt{100}.$

$-9$ $-10$

## Identify integers, rational numbers, irrational numbers, and real numbers

We have already described numbers as counting number s , whole number s , and integers    . What is the difference between these types of numbers?

$\begin{array}{cccccc}\text{Counting numbers}\hfill & & & & & 1,2,3,4,\text{…}\hfill \\ \text{Whole numbers}\hfill & & & & & 0,1,2,3,4,\text{…}\hfill \\ \text{Integers}\hfill & & & & & \text{…}-3,-2,-1,0,1,2,3,\text{…}\hfill \end{array}$

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number    is a number that can be written as a ratio of two integers.

## Rational number

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

A rational number can be written as the ratio of two integers.

All signed fractions, such as $\frac{4}{5},-\phantom{\rule{0.2em}{0ex}}\frac{7}{8},\frac{13}{4},-\phantom{\rule{0.2em}{0ex}}\frac{20}{3}$ are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to $\frac{3}{1},\frac{6}{2},\frac{9}{3},\frac{12}{4},\frac{15}{5}\text{…}$

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

$\begin{array}{ccccccc}\hfill 3=\frac{3}{1}\hfill & & & \hfill -8=-\phantom{\rule{0.2em}{0ex}}\frac{8}{1}\hfill & & & \hfill 0=\frac{0}{1}\hfill \end{array}$

Since any integer can be written as the ratio of two integers, all integers are rational numbers ! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer $-8$ could be written as the decimal $-8.0.$ So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means $7\frac{3}{10},$ we can write it as an improper fraction, $\frac{73}{10}.$ So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or $-1.2684\right)$ is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Write as the ratio of two integers: $-27$ 7.31.

$\begin{array}{cccccc}& & & & & -27\hfill \\ \\ \\ \text{Write it as a fraction with denominator}\phantom{\rule{0.2em}{0ex}}1.\hfill & & & & & \frac{-27}{1}\hfill \end{array}$

$\begin{array}{cccccc}& & & & & \phantom{\rule{0.3em}{0ex}}7.31\hfill \\ \\ \\ \begin{array}{c}\text{Write is as a mixed number. Remember.}\hfill \\ 7\phantom{\rule{0.2em}{0ex}}\text{is the whole number and the decimal}\hfill \\ \text{part,}\phantom{\rule{0.2em}{0ex}}0.31,\phantom{\rule{0.2em}{0ex}}\text{indicates hundredths.}\hfill \end{array}\hfill & & & & & \phantom{\rule{0.3em}{0ex}}7\frac{31}{100}\hfill \\ \\ \\ \text{Convert to an improper fraction.}\hfill & & & & & \phantom{\rule{0.3em}{0ex}}\frac{731}{100}\hfill \end{array}$

So we see that $-27$ and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Write as the ratio of two integers: $-24$ 3.57.

$\frac{-24}{1}$ $\frac{357}{100}$

Write as the ratio of two integers: $-19$ 8.41.

$\frac{-19}{1}$ $\frac{841}{100}$

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number , since $a=\frac{a}{1}$ for any integer, a . We can also change any integer to a decimal by adding a decimal point and a zero.

$\begin{array}{ccccccccccccccccccccc}\text{Integer}\hfill & & & & & -2\hfill & & & -1\hfill & & & 0\hfill & & & 1\hfill & & & 2\hfill & & & 3\hfill \\ \text{Decimal form}\hfill & & & & & -2.0\hfill & & & -1.0\hfill & & & 0.0\hfill & & & 1.0\hfill & & & 2.0\hfill & & & 3.0\hfill \end{array}$
$\text{These decimal numbers stop.}$

We have also seen that every fraction is a rational number . Look at the decimal form of the fractions we considered above.

$\begin{array}{ccccccccccccccc}\text{Ratio of integers}\hfill & & & & & \frac{4}{5}\hfill & & & -\phantom{\rule{0.2em}{0ex}}\frac{7}{8}\hfill & & & \frac{13}{4}\hfill & & & -\phantom{\rule{0.2em}{0ex}}\frac{20}{3}\hfill \\ \text{The decimal form}\hfill & & & & & 0.8\hfill & & & -0.875\hfill & & & 3.25\hfill & & & \begin{array}{}\\ -6.666\text{…}\hfill \\ -6.\stackrel{\text{–}}{6}\hfill \end{array}\hfill \end{array}$

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?
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Angelita
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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
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
hi
Darianna
hello
Mister
Eric here
Eric
6
Cindy