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This figure consists of a Venn diagram. To start there is a large rectangle marked Real Numbers. The right half of the rectangle consists of Irrational Numbers. The left half consists of Rational Numbers. Within the Rational Numbers rectangle, there are Integers …, negative 2, negative 1, 0, 1, 2, …. Within the Integers rectangle, there are Whole Numbers 0, 1, 2, 3, … Within the Whole Numbers rectangle, there are Counting Numbers 1, 2, 3, …
This chart shows the number sets that make up the set of real numbers. Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be?

Can we simplify −25 ? Is there a number whose square is −25 ?

( ) 2 = −25 ?

None of the numbers that we have dealt with so far has a square that is −25 . Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to −25 .

The square root of a negative number is not a real number.

For each number given, identify whether it is a real number or not a real number: −169 64 .

  1. There is no real number whose square is −169 . Therefore, −169 is not a real number.
  2. Since the negative is in front of the radical, 64 is −8 , Since −8 is a real number, 64 is a real number.
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For each number given, identify whether it is a real number or not a real number: −196 81 .

not a real number real number

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For each number given, identify whether it is a real number or not a real number: 49 −121 .

real number not a real number

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Given the numbers −7 , 14 5 , 8 , 5 , 5.9 , 64 , list the whole numbers integers rational numbers irrational numbers real numbers.

Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and −7 is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so 64 = −8 . So the integers are −7 , 8 , 64 .
Since all integers are rational, then −7 , 8 , 64 are rational. Rational numbers also include fractions and decimals that repeat or stop, so 14 5 and 5.9 are rational. So the list of rational numbers is −7 , 14 5 , 8 , 5.9 , 64 .
Remember that 5 is not a perfect square, so 5 is irrational.
All the numbers listed are real numbers.

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For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: −3 , 2 , 0. 3 , 9 5 , 4 , 49 .

4 , 49 −3 , 4 , 49 −3 , 0. 3 , 9 5 , 4 , 49 2 −3 , 2 , 0. 3 , 9 5 , 4 , 49

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For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: 25 , 3 8 , −1 , 6 , 121 , 2.041975

6 , 121 25 , −1 , 6 , 121 25 , 3 8 , −1 , 6 , 121 2.041975 25 , 3 8 , −1 , 6 , 121 , 2.041975

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Locate fractions on the number line

The last time we looked at the number line    , it only had positive and negative integers on it. We now want to include fraction    s and decimals on it.

Doing the Manipulative Mathematics activity “Number Line Part 3” will help you develop a better understanding of the location of fractions on the number line.

Let’s start with fractions and locate 1 5 , 4 5 , 3 , 7 4 , 9 2 , −5 , and 8 3 on the number line.

We’ll start with the whole numbers 3 and −5 . because they are the easiest to plot. See [link] .

The proper fractions listed are 1 5 and 4 5 . We know the proper fraction 1 5 has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts 1 5 , 2 5 , 3 5 , 4 5 . We plot 1 5 . See [link] .

Similarly, 4 5 is between 0 and −1 . After dividing the unit into 5 equal parts we plot 4 5 . See [link] .

Finally, look at the improper fractions 7 4 , 9 2 , 8 3 . These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See [link] .

Questions & Answers

What is the lcm of 340
Kendra Reply
How many numbers each equal to y must be taken to make 15xy
Malik Reply
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
Adda Reply
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Adda
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
Adda
(×+1)(×-4) = x^2-3×-4
Gee
Quadratic equations involving factorization
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5x + 1/3= 2x + 1/2
sanam
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5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
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sanam
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
vinaya Reply
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs $4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs $5.71 per gallon?
Mohamed Reply
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
muhammad Reply
4x-7y=8 2x-7y=1 what is the answer?
Ramil Reply
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
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usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
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copy and complete the table. x. 5. 8. 12. then 9x-5. to the 2nd power+4. then 2xto the second power +3x
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What is c+4=8
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2
Letha
4
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4
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4
thinking
C+4=8 -4 -4 C =4
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4+4=8
William
During two years in college, a student earned $9,500. The second year, she earned $500 more than twice the amount she earned the first year.
Nicole Reply
9500=500+2x
Debra
9500-500=9000 9000÷2×=4500 X=4500
Debra
X + Y = 9500....... & Y = 500 + 2X so.... X + 500 + 2X = 9500, them X = 3000 & Y = 6500
Pbp
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.
Josh Reply
Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Ida Reply
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Juan
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patrick
The sum of two numbers is 155. The difference is 23. Find the numbers
Michelle Reply
The sum of two numbers is 155. Their difference is 23. Find the numbers
Michelle
The difference between 89 and 66 is 23
Ciid
Joy is preparing 20 liters of a 25% saline solution. She only has 40% and 10% solution in her lab. How many liters of the 40% and how many liters of the 10% should she mix to make the 25% solution?
Amber Reply
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bismark
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Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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