Practice Real Numbers in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The complete set of all rational and irrational numbers, filling every point on the continuous number line.

Any number you can point to on an infinitely precise number line.

Showing a random 20 of 50 problems.

Example 1

easy
Give a real number that is neither rational nor an integer.

Example 2

challenge
Are there real numbers a,ba, b, both irrational, such that aba^b is rational?

Example 3

medium
Write the repeating decimal 0.6‾0.\overline{6} as a fraction to show it is rational.

Example 4

medium
Is 3⋅3\sqrt{3} \cdot \sqrt{3} rational or irrational?

Example 5

easy
Is the number 7 a real number?

Example 6

easy
Is 2\sqrt{2} a real number?

Example 7

easy
Is the decimal 0.1010010001…0.1010010001\ldots (one more 00 each time) rational?

Example 8

challenge
Is there a real number xx with x2=−4x^2 = -4? Explain.

Example 9

hard
What kind of decimal expansion does any rational number have?

Example 10

easy
Classify 16\sqrt{16} as rational or irrational.

Example 11

hard
Express 2.36‾2.\overline{36} as a fraction in lowest terms.

Example 12

challenge
Find a real number strictly between 13\frac{1}{3} and 12\frac{1}{2}.

Example 13

easy
Is −1\sqrt{-1} a real number?

Example 14

medium
Is the sum 2+(−2)\sqrt{2} + (-\sqrt{2}) rational or irrational?

Example 15

medium
Is 16\sqrt{16} rational or irrational?

Example 16

medium
Classify each as rational or irrational: 58\frac{5}{8}, 3\sqrt{3}.

Example 17

medium
Is 2+2\sqrt{2} + \sqrt{2} rational or irrational?

Example 18

challenge
Show that 2+3\sqrt{2} + \sqrt{3} is irrational.

Example 19

medium
Which subset does −49-\sqrt{49} belong to: integers, rationals, irrationals?

Example 20

medium
Is 00 a real number, and is it rational?
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