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Identify each of the following as rational or irrational:

0.29 0.81 6 2.515115111…

  1. rational
  2. rational
  3. irrational
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Identify each of the following as rational or irrational:

0.2 3 0.125 0.418302…

  1. rational
  2. rational
  3. irrational

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Let's think about square roots now. Square roots of perfect squares are always whole numbers    , so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Identify each of the following as rational or irrational:

36

44

Solution

The number 36 is a perfect square, since 6 2 = 36 . So 36 = 6 . Therefore 36 is rational.

Remember that 6 2 = 36 and 7 2 = 49 , so 44 is not a perfect square.

This means 44 is irrational.

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Identify each of the following as rational or irrational:

81

17

  1. rational
  2. irrational
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Identify each of the following as rational or irrational:

116

121

  1. irrational
  2. rational
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Classify real numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers    are rational numbers . Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers , we get the set of real numbers .

[link] illustrates how the number sets are related.

The image shows a large rectangle labeled “Real Numbers”. The rectangle is split in half vertically. The right half is labeled “Irrational Numbers”. The left half is labeled “Rational Numbers” and contains three concentric rectangles. The outer most rectangle is labeled “Integers”, the next rectangle is “Whole Numbers” and the inner most rectangle is “Natural Numbers”.
This diagram illustrates the relationships between the different types of real numbers.

Real numbers

Real numbers are numbers that are either rational or irrational.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Determine whether each of the numbers in the following list is a whole number, integer, rational number, irrational number, and real number.

−7 , 14 5 , 8 , 5 , 5.9 , 64

Solution

The whole numbers are 0 , 1 , 2 , 3 ,… The number 8 is the only whole number given.

The integers are the whole numbers, their opposites, and 0 . From the given numbers, −7 and 8 are integers. Also, notice that 64 is the square of 8 so 64 = −8 . So the integers are −7 , 8 , 64 .

Since all integers are rational, the numbers −7 , 8 , and 64 are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so 14 5 and 5.9 are rational.

The number 5 is not a perfect square, so 5 is irrational.

All of the numbers listed are real.

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We'll summarize the results in a table.

Number Whole Integer Rational Irrational Real
−7
14 5
8
5
5.9
64
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Determine whether each number is a whole number, integer, rational number, irrational number, and real number: −3 , 2 , 0 . 3 , 9 5 , 4 , 49 .


The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled “Number”, the second column “Whole”, the third “Integer”, the fourth “Rational” the fifth “Irrational” and the sixth “Real”. Each row has a number in the “Number” column then an x in each column that corresponds to the type of number it is. The second row has the number negative 3 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The third row has the number negative square root of 2 in the “Number” column and an x marked in the “Irrational” and “Real” columns. The fourth row has the number 0.3 repeating in the “Number” column and an x marked in the “Rational” and “Real” columns. The fifth row has the number  square root of negative 49 in the “Number” column with no other columns marked. The sixth row has the number 4 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns.  The last row has the number 9 fifths in the “Number” column and an x marked in the “Rational” and “Real” columns.

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Determine whether each number is a whole number, integer, rational number, irrational number, and real number: 25 , 3 8 , −1 , 6 , 121 , 2.041975…


The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled “Number”, the second column “Whole”, the third “Integer”, the fourth “Rational” the fifth “Irrational” and the sixth “Real”. Each row has a number in the “Number” column then an x in each column that corresponds to the type of number it is. The second row has the number negative square root of 25 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The third row has the number negative 3 eights in the “Number” column and an x marked in the “Rational” and “Real” columns. The fourth row has the number negative 1 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The fifth row has the number 6 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns. The sixth row has the number square root of 121 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns. The last row has the number 2.041975 followed by an ellipsis in the “Number” column and an x marked in the “Irrational” and “Real” columns.

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Key concepts

  • Real numbers
    • The image shows a large rectangle labeled “Real Numbers”. The rectangle is split in half vertically. The right half is labeled “Irrational Numbers”. The left half is labeled “Rational Numbers” and contains three concentric rectangles. The outer most rectangle is labeled “Integers”, the next rectangle is “Whole Numbers” and the inner most rectangle is “Natural Numbers”.

Practice makes perfect

Rational Numbers

In the following exercises, write as the ratio of two integers.

  1. 5
  2. 3.19
  1. 5 1
  2. 319 100
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  1. −12
  2. 9.279
  1. −12 1
  2. 9279 1000
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In the following exercises, determine which of the given numbers are rational and which are irrational.

0.75 , 0.22 3 , 1.39174…

Rational: 0.75 , 0.22 3 . Irrational: 1.39174…

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0.36 , 0.94729… , 2.52 8

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0 . 45 , 1.919293… , 3.59

Rational: 0 . 45 , 3.59 . Irrational: 1.919293…

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0.1 3 , 0.42982… , 1.875

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In the following exercises, identify whether each number is rational or irrational.

  1. 25
  2. 30
  1. rational
  2. irrational
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  1. 164
  2. 169
  1. irrational
  2. rational
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Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

−8 , 0 , 1.95286.... , 12 5 , 36 , 9


The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled “Number”, the second column “Whole”, the third “Integer”, the fourth “Rational” the fifth “Irrational” and the sixth “Real”. Each row has a number in the “Number” column then an x in each column that corresponds to the type of number it is. The second row has the number negative 8 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The third row has the number 0 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns. The fourth row has the number 1.95286 followed by and ellipsis in the “Number” column and an x marked in the “Irrational” and “Real” columns. The fifth row has the number 12 fifths in the “Number” column and an x marked in the “Rational” and “Real” columns. The sixth row has the number square root of 36 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns. The last row has the number 9 in the “Number” column and an x marked in the “Whole”, “Integer”, “Rational” and “Real” columns.

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−9 , −3 4 9 , 9 , 0.4 09 , 11 6 , 7

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100 , −7 , 8 3 , −1 , 0.77 , 3 1 4


The table has seven rows and six columns. The first row is a header row that labels each column. The first column is labeled “Number”, the second column “Whole”, the third “Integer”, the fourth “Rational” the fifth “Irrational” and the sixth “Real”. Each row has a number in the “Number” column then an x in each column that corresponds to the type of number it is. The second row has the number negative 100 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The third row has the number negative 7 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The fourth row has the number negative 8 thirds in the “Number” column and an x marked in the “Rational” and “Real” columns. The fifth row has the number negative 1 in the “Number” column and an x marked in the “Integer”, “Rational” and “Real” columns. The sixth row has the number 0.77 in the “Number” column and an x marked in the “Rational” and “Real” columns. The last row has the number 3 and 1 quarter in the “Number” column and an x marked in the “Rational” and “Real” columns.

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Everyday math

Field trip All the 5 th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

How many buses will be needed?

Why must the answer be a whole number?

Why shouldn't you round the answer the usual way?

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Child care Serena wants to open a licensed child care center. Her state requires that there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

How many teachers will be needed?

Why must the answer be a whole number?

Why shouldn't you round the answer the usual way?

  1. 4
  2. Teachers cannot be divided
  3. It would result in a lower number.

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Writing exercises

In your own words, explain the difference between a rational number and an irrational number.

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Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

Answers will vary.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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