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Real Numbers and Irrational Numbers Practice Tests

These Grade 8 Math practice tests cover real numbers, rational and irrational numbers, number lines, square roots, cube roots, exact radical forms, comparisons, approximations, and applications of the Pythagorean theorem.

Skills covered

  • Classifying natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers
  • Estimating and comparing square roots and cube roots using perfect powers and number-line reasoning
  • Simplifying radical expressions and preserving exact values with LaTeX notation
  • Using real numbers in distance, area, diagonal, circumference, and right-triangle problems
  • Distinguishing exact forms from rounded decimal approximations

How to use these practice tests

Classify each number first, then choose an exact form or estimate as requested. When comparing radicals, use nearby perfect squares or a calculator estimate and check the sign.

For more Grade 8 Math practice, return to the Grade 8 Math category.

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1. Test

Grade 8 Real Numbers and Irrational Numbers Practice Tests

Practice Grade 8 Math questions about real numbers, rational and irrational numbers, square roots, cube roots, exact forms, comparisons, and approximations.

Choose the best answer for each question. Use exact radical forms when requested and review the explanation after submitting your answers.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

Which statement defines a real number?

Please select an answer first.

Correct answer: A

The correct answer is A number that can be represented on the number line, including both rational and irrational numbers..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The real-number system includes rational numbers such as 34\frac{3}{4} and irrational numbers such as 2\sqrt{2}.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

The real-number system includes rational numbers such as 34\frac{3}{4} and irrational numbers such as 2\sqrt{2}.

2. QuestionMultiple choice

Which statement defines an irrational number?

Please select an answer first.

Correct answer: B

The correct answer is A real number that cannot be written as a ratio of two integers..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: An irrational number has a decimal expansion that is nonterminating and nonrepeating, so it cannot be expressed as ab\frac{a}{b} with integers aa and b≠0b\ne0.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

An irrational number has a decimal expansion that is nonterminating and nonrepeating, so it cannot be expressed as ab\frac{a}{b} with integers aa and b≠0b\ne0.

3. QuestionMultiple choice

Which number is rational?

Please select an answer first.

Correct answer: C

The correct answer is 0.1250.125.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 0.125=1251000=180.125=\frac{125}{1000}=\frac{1}{8}, so the decimal can be written as a ratio of integers.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

0.125=1251000=180.125=\frac{125}{1000}=\frac{1}{8}, so the decimal can be written as a ratio of integers.

4. QuestionMultiple choice

Which number is rational?

Please select an answer first.

Correct answer: D

The correct answer is 49\sqrt{49}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 49=7\sqrt{49}=7, and every integer is rational because it can be written over 1.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

49=7\sqrt{49}=7, and every integer is rational because it can be written over 1.

5. QuestionMultiple choice

Which number is irrational?

Please select an answer first.

Correct answer: A

The correct answer is 2\sqrt{2}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 2\sqrt{2} cannot be written as a ratio of integers, so its decimal expansion is nonterminating and nonrepeating.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

2\sqrt{2} cannot be written as a ratio of integers, so its decimal expansion is nonterminating and nonrepeating.

6. QuestionMultiple choice

Why is a terminating decimal rational?

Please select an answer first.

Correct answer: B

The correct answer is It can be written as a fraction with a power of 10 as the denominator and then simplified..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 0.37=371000.37=\frac{37}{100}, so a terminating decimal is a ratio of integers.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

0.37=371000.37=\frac{37}{100}, so a terminating decimal is a ratio of integers.

7. QuestionMultiple choice

Which type of decimal represents a rational number?

Please select an answer first.

Correct answer: C

The correct answer is A repeating decimal such as 0.6‾0.\overline{6}.

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: 0.6‾=230.\overline{6}=\frac{2}{3}, so a repeating decimal is rational.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

0.6‾=230.\overline{6}=\frac{2}{3}, so a repeating decimal is rational.

8. QuestionMultiple choice

Which statement about π\pi is true?

Please select an answer first.

Correct answer: D

The correct answer is It is irrational because its decimal expansion is nonterminating and nonrepeating..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: π\pi is a real irrational number commonly approximated by decimals such as 3.143.14.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

π\pi is a real irrational number commonly approximated by decimals such as 3.143.14.

9. QuestionMultiple choice

Why is every integer a rational number?

Please select an answer first.

Correct answer: A

The correct answer is Any integer nn can be written as n1\frac{n}{1}..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The denominator 1 is a nonzero integer, so every integer fits the definition of a rational number.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

The denominator 1 is a nonzero integer, so every integer fits the definition of a rational number.

10. QuestionMultiple choice

What does every point on the real number line represent?

Please select an answer first.

Correct answer: B

The correct answer is A real number, which may be rational or irrational..

  1. Classify the number: Determine whether the value is natural, whole, integer, rational, irrational, or real.
  2. Use an exact representation: The real number line includes negative numbers, zero, positive numbers, rational numbers, and irrational numbers.
  3. Check the approximation: If a decimal estimate is used, compare it with nearby perfect squares or known values and keep the required precision.

The real number line includes negative numbers, zero, positive numbers, rational numbers, and irrational numbers.

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