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Rational and Irrational Numbers

Identify rational numbers and irrational numbers

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

counting numbers\(1,2,3,\text{4\ldots }\)
whole numbers\(0,1,2,3,\text{4\ldots }\)
integers\(\text{\ldots }-3,-2,-1,0,1,2,3,\text{4\ldots }\)

Condensed — the full section is in OpenStax Prealgebra 2e.

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.

illustrates how the number sets are related.

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.

Example

Try it.

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

\[-7,\frac{14}{5},8,\sqrt{5},5.9,-\sqrt{64}\]
Solution

ⓐ The whole numbers are \(0,1,2,3\text{,\ldots }\) 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 \(-\sqrt{64}=-8.\) So the integers are \(-7,8,-\sqrt{64}.\)

ⓒ Since all integers are rational, the numbers \(-7,8,\text{and}\ -\sqrt{64}\) are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so \(\frac{14}{5}\ \text{and}\ 5.9\) are rational.

ⓓ The number \(5\) is not a perfect square, so \(\sqrt{5}\) is irrational.

ⓔ All of the numbers listed are real.

We'll summarize the results in a table.

NumberWholeIntegerRationalIrrationalReal
\(-7\)\(✓\)\(✓\)\(✓\)
\(\frac{14}{5}\)\(✓\)\(✓\)
\(8\)\(✓\)\(✓\)\(✓\)\(✓\)
\(\sqrt{5}\)\(✓\)\(✓\)
\(5.9\)\(✓\)\(✓\)
\(-\sqrt{64}\)\(✓\)\(✓\)\(✓\)

Rational and Irrational Numbers

Rational Numbers

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

Try it.

  1. ⓐ \(\ 5\)
  2. ⓑ \(\ 3.19\)
Solution
  1. ⓐ \(\ \frac{5}{1}\)
  2. ⓑ \(\ \frac{319}{100}\)

Try it.

  1. ⓐ \(\ 8\)
  2. ⓑ \(\ -1.61\)

Try it.

  1. ⓐ \(\ -12\)
  2. ⓑ \(\ 9.279\)
Solution
  1. ⓐ \(\ \frac{-12}{1}\)
  2. ⓑ \(\frac{9279}{1000}\)

Try it.

  1. ⓐ \(\ -16\\)
  2. ⓑ \(\ 4.399\)

In the following exercises, determine which of the given numbers are rational and which are irrational.

Try it.

\(0.75\), \(0.22\overset{\text{-}}{3}\), \(\text{1.39174\ldots }\)

Solution

Rational: \(0.75,0.22\overset{\text{-}}{3}\). Irrational: \(\text{1.39174\ldots }\)

Try it.

\(0.36\), \(\text{0.94729\ldots }\), \(2.52\overset{\text{-}}{8}\)

Try it.

\(0.\overset{\text{—}}{45}\), \(\text{1.919293\ldots }\), \(3.59\)

Solution

Rational: \(0.\overset{\text{—}}{45}\), \(3.59\). Irrational: \(\text{1.919293\ldots }\)

Try it.

\(0.1\overset{\text{-}}{3},\text{0.42982\ldots }\), \(1.875\)

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

Try it.

  1. ⓐ \(\ \sqrt{25}\)
  2. ⓑ \(\ \sqrt{30}\)
Solution
  1. ⓐ rational
  2. ⓑ irrational

Try it.

  1. ⓐ \(\ \sqrt{44}\\)
  2. ⓑ \(\sqrt{49}\)

Try it.

  1. ⓐ \(\ \sqrt{164}\)
  2. ⓑ \(\ \sqrt{169}\)
Solution
  1. ⓐ irrational
  2. ⓑ rational

Try it.

  1. ⓐ \(\ \sqrt{225}\)
  2. ⓑ \(\ \sqrt{216}\)

Classifying Real Numbers

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

Try it.

\(-8\), \(0,\text{1.95286....}\), \(\frac{12}{5}\), \(\sqrt{36}\), \(9\)

Solution


Try it.

\(-9\), \(-3\frac{4}{9}\), \(-\sqrt{9}\), \(0.4\overset{\text{—}}{09}\),\(\frac{11}{6}\), \(7\)

Try it.

\(-\sqrt{100}\), \(-7\), \(-\frac{8}{3}\), \(-1\), \(0.77\), \(3\frac{1}{4}\)

Solution


Try it.

Field trip All the \(5\text{th}\) 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?

Try it.

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.

  1. ⓐ How many teachers will be needed?

  2. ⓑ Why must the answer be a whole number?

  3. ⓒ Why shouldn't you round the answer the usual way?

Solution

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

Try it.

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

Try it.

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

Solution

Answers will vary.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (34)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Write \(3.19\) as an improper fraction.
    If you missed this problem, review .

    Одкриј го одговорот

    \(\frac{319}{100}\)

  2. Write \(\frac{5}{11}\) as a decimal.
    If you missed this problem, review .

    Одкриј го одговорот

    \(0.\overset{\bar}{45}\)

  3. Simplify: \(\sqrt{144}.\)
    If you missed this problem, review .

    Одкриј го одговорот

    \(12\)

  4. Write each as the ratio of two integers: ⓐ \(-15\)ⓑ \(\ 6.81\)ⓒ \(\ -3\frac{6}{7}.\)

    Одкриј го одговорот
    ⓐ
    \(-15\)
    Write the integer as a fraction with denominator 1.\(\frac{-15}{1}\)
    ⓑ
    \(6.81\)
    Write the decimal as a mixed number.\(6\frac{81}{100}\)
    Then convert it to an improper fraction.\(\frac{681}{100}\)
    ⓒ
    \(-3\frac{6}{7}\)
    Convert the mixed number to an improper fraction.\(-\frac{27}{7}\)

  5. Write each as the ratio of two integers: ⓐ \(\ -24\)ⓑ \(\ 3.57.\)

    Одкриј го одговорот
    1. ⓐ \(\ \frac{-24}{1}\)
    2. ⓑ \(\ \frac{357}{100}\)

  6. Write each as the ratio of two integers: ⓐ \(\ -19\) ⓑ \(\ 8.41.\)

    Одкриј го одговорот
    1. ⓐ \(\ \frac{-19}{1}\\)
    2. ⓑ \(\ \frac{841}{100}\)

  7. Identify each of the following as rational or irrational:

    1. ⓐ \(\ 0.58\overset{\text{-}}{3}\)
    2. ⓑ \(\ 0.475\)
    3. ⓒ \(\ \text{3.605551275\ldots }\)
    Одкриј го одговорот

    ⓐ \(0.58\overset{\text{-}}{3}\)
    The bar above the \(3\) indicates that it repeats. Therefore, \(0.58\overset{\text{-}}{3}\) is a repeating decimal, and is therefore a rational number.

    ⓑ \(0.475\)
    This decimal stops after the \(5\), so it is a rational number.

    ⓒ \(\text{3.605551275\ldots }\)
    The ellipsis \(\text{(\ldots )}\) means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.

  8. Identify each of the following as rational or irrational:

    ⓐ \(\ 0.29\)ⓑ \(\ 0.81\overset{\text{-}}{6}\\)ⓒ \(\ \text{2.515115111\ldots }\)

    Одкриј го одговорот
    1. ⓐ rational
    2. ⓑ rational
    3. ⓒ irrational

  9. Identify each of the following as rational or irrational:

    ⓐ \(\ 0.2\overset{\text{-}}{3}\)ⓑ \(\ 0.125\)ⓒ \(\ \text{0.418302\ldots }\)

    Одкриј го одговорот

    1. ⓐ rational
    2. ⓑ rational
    3. ⓒ irrational

  10. Identify each of the following as rational or irrational:

    1. ⓐ \(\ \sqrt{36}\)

    2. ⓑ \(\ \sqrt{44}\)

    Одкриј го одговорот

    ⓐ The number \(36\) is a perfect square, since \({6}^{2}=36.\) So \(\sqrt{36}=6.\) Therefore \(\sqrt{36}\) is rational.

    ⓑ Remember that \({6}^{2}=36\) and \({7}^{2}=49,\) so \(44\) is not a perfect square.

    This means \(\sqrt{44}\) is irrational.

  11. Identify each of the following as rational or irrational:

    1. ⓐ \(\ \sqrt{81}\)
    2. ⓑ \(\ \sqrt{17}\)
    Одкриј го одговорот
    1. ⓐ rational
    2. ⓑ irrational

  12. Identify each of the following as rational or irrational:

    1. ⓐ \(\ \sqrt{116}\)

    2. ⓑ \(\ \sqrt{121}\)

    Одкриј го одговорот
    1. ⓐ irrational
    2. ⓑ rational

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

    \[-7,\frac{14}{5},8,\sqrt{5},5.9,-\sqrt{64}\]
    Одкриј го одговорот

    ⓐ The whole numbers are \(0,1,2,3\text{,\ldots }\) 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 \(-\sqrt{64}=-8.\) So the integers are \(-7,8,-\sqrt{64}.\)

    ⓒ Since all integers are rational, the numbers \(-7,8,\text{and}\ -\sqrt{64}\) are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so \(\frac{14}{5}\ \text{and}\ 5.9\) are rational.

    ⓓ The number \(5\) is not a perfect square, so \(\sqrt{5}\) is irrational.

    ⓔ All of the numbers listed are real.

    We'll summarize the results in a table.

    NumberWholeIntegerRationalIrrationalReal
    \(-7\)\(✓\)\(✓\)\(✓\)
    \(\frac{14}{5}\)\(✓\)\(✓\)
    \(8\)\(✓\)\(✓\)\(✓\)\(✓\)
    \(\sqrt{5}\)\(✓\)\(✓\)
    \(5.9\)\(✓\)\(✓\)
    \(-\sqrt{64}\)\(✓\)\(✓\)\(✓\)

  14. Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: \(-3,-\sqrt{2},0.\overset{\text{-}}{3},\frac{9}{5},4,\sqrt{49}.\)

    Одкриј го одговорот


  15. Determine whether each number is a ⓐ whole number, ⓑ integer, ⓒ rational number, ⓓ irrational number, and ⓔ real number: \(-\sqrt{25},-\frac{3}{8},-1,6,\sqrt{121},\text{2.041975\ldots }\)

    Одкриј го одговорот


    1. ⓐ \(\ 5\)
    2. ⓑ \(\ 3.19\)
    Одкриј го одговорот
    1. ⓐ \(\ \frac{5}{1}\)
    2. ⓑ \(\ \frac{319}{100}\)
    1. ⓐ \(\ 8\)
    2. ⓑ \(\ -1.61\)
    1. ⓐ \(\ -12\)
    2. ⓑ \(\ 9.279\)
    Одкриј го одговорот
    1. ⓐ \(\ \frac{-12}{1}\)
    2. ⓑ \(\frac{9279}{1000}\)
    1. ⓐ \(\ -16\\)
    2. ⓑ \(\ 4.399\)

  20. \(0.75\), \(0.22\overset{\text{-}}{3}\), \(\text{1.39174\ldots }\)

    Одкриј го одговорот

    Rational: \(0.75,0.22\overset{\text{-}}{3}\). Irrational: \(\text{1.39174\ldots }\)

  21. \(0.36\), \(\text{0.94729\ldots }\), \(2.52\overset{\text{-}}{8}\)

  22. \(0.\overset{\text{—}}{45}\), \(\text{1.919293\ldots }\), \(3.59\)

    Одкриј го одговорот

    Rational: \(0.\overset{\text{—}}{45}\), \(3.59\). Irrational: \(\text{1.919293\ldots }\)

  23. \(0.1\overset{\text{-}}{3},\text{0.42982\ldots }\), \(1.875\)

    1. ⓐ \(\ \sqrt{25}\)
    2. ⓑ \(\ \sqrt{30}\)
    Одкриј го одговорот
    1. ⓐ rational
    2. ⓑ irrational
    1. ⓐ \(\ \sqrt{44}\\)
    2. ⓑ \(\sqrt{49}\)
    1. ⓐ \(\ \sqrt{164}\)
    2. ⓑ \(\ \sqrt{169}\)
    Одкриј го одговорот
    1. ⓐ irrational
    2. ⓑ rational
    1. ⓐ \(\ \sqrt{225}\)
    2. ⓑ \(\ \sqrt{216}\)

  28. \(-8\), \(0,\text{1.95286....}\), \(\frac{12}{5}\), \(\sqrt{36}\), \(9\)

    Одкриј го одговорот


  29. \(-9\), \(-3\frac{4}{9}\), \(-\sqrt{9}\), \(0.4\overset{\text{—}}{09}\),\(\frac{11}{6}\), \(7\)

  30. \(-\sqrt{100}\), \(-7\), \(-\frac{8}{3}\), \(-1\), \(0.77\), \(3\frac{1}{4}\)

    Одкриј го одговорот


  31. Field trip All the \(5\text{th}\) 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?

  32. 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.

    1. ⓐ How many teachers will be needed?

    2. ⓑ Why must the answer be a whole number?

    3. ⓒ Why shouldn't you round the answer the usual way?

    Одкриј го одговорот

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

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

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

    Одкриј го одговорот

    Answers will vary.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\approx
approximately equal
Equal to the precision shown, not exactly.
a \bmod n
remainder
What is left after dividing a by n.
\%
per cent
Per hundred: 15% = 15/100.
a : b,\ \frac{a}{b}
ratio, fraction
a for every b; a divided by b.

How to: Rational and Irrational Numbers

  1. Identify rational numbers and irrational numbers
  2. Classify different types of real numbers
  3. stops or repeats, the number is rational.
  4. does not stop and does not repeat, the number is irrational.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Обиди се со себе.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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