Irrational Numbers Examples in Math

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

An irrational number is a real number that cannot be expressed as a ratio of two integers $\frac{p}{q}$ ; its decimal expansion goes on forever without repeating any fixed block of digits.

$\pi$ and $\sqrt{2}$ go on forever without any pattern—you can't write them as a fraction.

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: Irrational numbers are exact numbers whose decimals never terminate or repeat.

Common stuck point: The procedure for irrational numbers is the easy part; the trap is thinking irrational means impossible or fake. Asking "Can this number be written exactly as a ratio of integers?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can this number be written exactly as a ratio of integers?

Worked Examples

Example 1

easy

Classify each number as rational or irrational:

\sqrt{16}

\sqrt{5}

0.75

\pi

Answer

\text{Rational: } \sqrt{16},\; 0.75 \qquad \text{Irrational: } \sqrt{5},\; \pi

First step

\sqrt{16} = 4

, which is an integer, so it is rational.

0.75 = \frac{3}{4}

, also rational.

Full solution

2
$\sqrt{5}$ is not a perfect square, so $\sqrt{5}$ is irrational. $\pi$ is a well-known irrational number.
3
Rational: $\sqrt{16}$ , $0.75$ . Irrational: $\sqrt{5}$ , $\pi$ .

A number is rational if it can be written as

\frac{a}{b}

with integers

a, b

(

b \neq 0

). Square roots of non-perfect-squares are irrational, and

\pi

cannot be expressed as any fraction.

Example 2

medium

Show that

\sqrt{2}

lies between

1.4

and

1.5

, then estimate it to one decimal place.

Example 3

medium

Prove that

\sqrt{3}

is irrational by contradiction.

Example 4

medium

Approximate

\sqrt{20}

to one decimal place without a calculator.

Practice Problems

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

Example 1

easy

\sqrt{49} + \sqrt{3}

rational or irrational? Explain.

Example 2

easy

3 + \sqrt{7}

rational or irrational?

Example 3

easy

\sqrt{2}

rational or irrational?

Example 4

easy

\pi

rational or irrational?

Example 5

easy

\sqrt{25}

rational or irrational?

Example 6

easy

Is the decimal

0.101001000\ldots

(one more zero each time) rational?

Example 7

easy

\sqrt{7}

irrational?

Example 8

easy

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

Example 9

easy

\sqrt{100}

rational or irrational?

Example 10

easy

Between which two consecutive integers does

\sqrt{10}

lie?

Example 11

medium

Classify each as rational or irrational:

\sqrt{36}

\sqrt{3}

\frac{\pi}{2}

Example 12

medium

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

rational or irrational?

Example 13

medium

\sqrt{2}+\sqrt{2}

rational or irrational?

Example 14

medium

Give an example of two irrational numbers whose sum is rational.

Example 15

medium

Simplify

\sqrt{50}

and state whether it is rational.

Example 16

medium

0.123456789101112\ldots

(concatenating the counting numbers) rational?

Example 17

medium

Order from least to greatest:

\sqrt{2},\ 1.5,\ \sqrt{3}

Example 18

challenge

Prove that

\sqrt{2}

is irrational (outline the contradiction).

Example 19

challenge

Is it possible for an irrational raised to an irrational power to be rational? Give an idea.

Example 20

challenge

Show that between

\sqrt{2}

and

\sqrt{3}

there is a rational number, and name one.

Example 21

medium

Rationalize and classify

\frac{1}{\sqrt{2}}

Example 22

medium

\frac{\pi}{\pi}

rational?

Example 23

easy

Between which two consecutive integers does

\sqrt{19}

lie?

Example 24

easy

True or false: every terminating decimal is rational.

Example 25

medium

\sqrt{2} \cdot \sqrt{18}

rational or irrational?

Example 26

medium

Classify each:

\sqrt{0.04}

\sqrt{0.5}

Example 27

medium

\sqrt{2} + \sqrt{3}

rational?

Example 28

medium

Give an example of an irrational number between

1.1

and

1.2

Example 29

medium

Simplify

\sqrt{72}

and classify it.

Example 30

medium

\sqrt{2} \cdot \pi

rational or irrational?

Example 31

medium

Order from least to greatest:

\sqrt{8}, 3, \pi

Example 32

medium

Give two irrationals whose product is rational.

Example 33

hard

Prove that

\sqrt{5}

is irrational.

Example 34

hard

\sqrt{2} + \sqrt{3} - \sqrt{5}

rational? Briefly justify.

Example 35

hard

r

is rational and

s

is irrational, must

r + s

be irrational?

Example 36

hard

Show that between any two distinct rationals there exists an irrational.

Example 37

hard

\log_{10} 2

rational or irrational?

Example 38

challenge

Provide an example showing an irrational raised to an irrational power can be rational.

Example 39

challenge

Show that

\sqrt{2} + \sqrt{3}

is a root of a polynomial with integer coefficients; what is its minimal polynomial?

← Back to Irrational Numbers Formula Explained Practice Mode

Related Concepts

Rational Numbers Square Roots Real Numbers Limit

Background Knowledge

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

rational numberssquare roots