Real Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Real Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The real numbers fill the line completely — all rationals plus all irrationals, with no gaps.

Common stuck point: The procedure for real numbers is the easy part; the trap is excluding irrationals from the reals. Asking "Can this value be located as a single point on the ordinary number line (no imaginary part)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can this value be located as a single point on the ordinary number line (no imaginary part)?

Worked Examples

Example 1

easy

Classify each number as rational or irrational, and state whether it is a real number:

-7

\sqrt{9}

\sqrt{7}

\pi

Answer

\text{All four are real numbers; } {-7,\,\sqrt{9}} \text{ rational; } {\sqrt{7},\,\pi} \text{ irrational}

First step

-7 = \frac{-7}{1}

is rational.

\sqrt{9} = 3 = \frac{3}{1}

is rational. Both are real.

Full solution

2
$\sqrt{7}$ is irrational (7 is not a perfect square); $\pi$ is irrational (proven). Both are real.
3
All four are real numbers: $\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$ .

Every rational and every irrational number is a real number. The real numbers form the complete number line with no gaps. Being 'real' does not mean 'rational' — most real numbers are in fact irrational.

Example 2

medium

Show that between any two distinct real numbers

a

and

b

(with

a < b

), there exists another real number.

Example 3

easy

Convert

0.\overline{6}

to a fraction.

Example 4

medium

Convert

0.\overline{12}

to a fraction in lowest terms.

Example 5

medium

True or false: the sum of a rational and an irrational number is always irrational.

Example 6

medium

Convert

0.4\overline{5}

to a fraction.

Example 7

hard

Prove that

\sqrt{2}

is irrational.

Example 8

hard

Show that if

r

is rational and

s

irrational, then

rs

is irrational whenever

r \ne 0

Example 9

hard

Express

2.\overline{36}

as a fraction in lowest terms.

Example 10

challenge

Show that

\sqrt{2} + \sqrt{3}

is irrational.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Which of the following are NOT real numbers?

\sqrt{-4}

-\sqrt{4}

\sqrt[3]{-8}

\frac{1}{0}

Example 2

medium

Give one example of a real number between

\sqrt{2}

and

\sqrt{3}

Example 3

easy

Is the number 7 a real number?

Example 4

easy

\frac{3}{4}

a real number?

Example 5

easy

\sqrt{2}

a real number?

Example 6

easy

-5

a real number?

Example 7

easy

Which set does

0.333\ldots

(repeating) belong to: rational or irrational?

Example 8

easy

Is the number

\pi

rational or irrational?

Example 9

easy

\sqrt{-1}

a real number?

Example 10

easy

True or false: every rational number is a real number.

Example 11

medium

Classify each as rational or irrational:

\frac{5}{8}

\sqrt{3}

Example 12

medium

\sqrt{16}

rational or irrational?

Example 13

medium

Between which two consecutive integers does

\sqrt{20}

lie?

Example 14

medium

Write the repeating decimal

0.\overline{6}

as a fraction to show it is rational.

Example 15

medium

Is the sum

\sqrt{2} + (-\sqrt{2})

rational or irrational?

Example 16

medium

Order on the number line:

\sqrt{2}

1.5

\frac{4}{3}

Example 17

medium

0

a real number, and is it rational?

Example 18

medium

Which is larger,

\sqrt{10}

3

Example 19

challenge

Prove

\sqrt{9}

is rational but

\sqrt{8}

is irrational, and state which is larger.

Example 20

challenge

Is there a real number

x

with

x^2 = -4

? Explain.

Example 21

challenge

Find a real number strictly between

\frac{1}{3}

and

\frac{1}{2}

Example 22

medium

Is the product

\sqrt{2}\cdot\sqrt{2}

rational or irrational?

Example 23

easy

Classify

\sqrt{16}

as rational or irrational.

Example 24

easy

Order from least to greatest:

-2.5, \sqrt{2}, \frac{3}{2}, -1

Example 25

easy

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

Example 26

easy

Is the decimal

0.1010010001\ldots

(one more

0

each time) rational?

Example 27

medium

Find a rational number between

\sqrt{5}

and

\sqrt{6}

Example 28

medium

Is the product

\sqrt{2} \cdot \sqrt{8}

rational or irrational?

Example 29

medium

Which subset does

-\sqrt{49}

belong to: integers, rationals, irrationals?

Example 30

medium

\sqrt{3} \cdot \sqrt{3}

rational or irrational?

Example 31

hard

\sqrt{2} \cdot \sqrt{3}

rational or irrational?

Example 32

hard

Give an irrational number between

\frac{1}{2}

and

\frac{3}{4}

Example 33

hard

Compute and classify:

(\sqrt{5} + 1)(\sqrt{5} - 1)

Example 34

challenge

Are there real numbers

a, b

, both irrational, such that

a^b

is rational?

← Back to Real Numbers Formula Explained Practice Mode

Related Concepts

Rational Numbers Irrational Numbers Complex Numbers Limit

Background Knowledge

These ideas may be useful before you work through the harder examples.

rational numbersirrational numbers