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.

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

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
whats the coefficient of 17x
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
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?
Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan has ridden 1.5 hours and Leo has ridden 2 hours, they meet on the path. Ethan’s speed is 6 miles per hour faster than Leo’s speed. Find the speed of the two bikers.
Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel?
Mckenzie
Nancy took a 3 hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
90 minutes