# 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.

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

rectangular field solutions
What is this?
Donna
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
?
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?
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 Reply p mulripied-5 and add30 Tausif p mulripied-5 and addto30 Tausif Can you explain further Monica Reply p mulripied-5 and add to 30 Tausif How do you find divisible numbers without a calculator? Jacob Reply 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? Ariana Reply 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? Kirisma Reply 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 hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused. Alicia Reply what do you need help in? Felix subtracting a negative....is adding!! 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 Niazmohammad 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
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
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul