Irrational Numbers - Math Steps, Examples & Questions

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Irrational numbers

Here you will learn about irrational numbers, including what an irrational number is, examples of irrational numbers and how to identify irrational numbers.

Students will first learn about irrational numbers as part of the number system in $8$ th grade.

What is an irrational number?

An irrational number is a real number or set of real numbers that cannot be written as a fraction of two integers (whole numbers). It is a non-terminating decimal that cannot be expressed as a fraction.

For example,

$\sqrt{3}=1.73205…$ is a non-terminating decimal number which is irrational because it cannot be expressed as a fraction in the form $\cfrac{a}{b}$ where $a$ and $b$ are integers.

Properties of irrational numbers

Irrational numbers have several properties that distinguish them from rational numbers. A set of irrational numbers can have the following properties:

The decimal representation, or decimal expansion of an irrational number continues on forever, without repeating.
Irrational numbers cannot be expressed in the form of a ratio of integers.
The square roots of non-perfect squares are always irrational.
The least common multiple (LCM) of any two irrational numbers may or may not exist.

Famous irrational numbers

There are several famous irrational numbers. These include,

Surds are types of irrational numbers. A surd is the root of a number which produces a non-terminating decimal.

For example,

\sqrt{9}=3 = \text{ rational}

\sqrt{2}=1.414… = \text{ surd } = \text{ irrational}

\sqrt{0.25} = \sqrt{\cfrac{25}{100}} = \cfrac{\sqrt{25}}{\sqrt{100}} = \cfrac{5}{10} = \cfrac{1}{2} = \text{ rational}

\sqrt[3]{64} = 4 = \text{ rational}

\sqrt[4]{30} =2.34034731932... = \text{ surd } = \text{ irrational}

What is an irrational number?

Common Core State Standards

How does this relate to $8$ th grade math?

Grade 8: The Number System (8.NS.A.1)
Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

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How to identify if a number is an irrational number

In order to identify if a number is an irrational number:

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.
Identify what type of root it is and write a list of corresponding powers.
Identify where the integer inside the root falls on the list.

Irrational numbers examples

Example 1: identifying if the root of an integer is rational or irrational without a calculator

Is $\sqrt{30}$ a rational or an irrational number?

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$30$ is an integer.

2Identify what type of root it is and write a list of corresponding powers.

This is a square root.

The list of square numbers is $1,4,9,16,25,36,49\dots$

3Identify where the integer inside the root falls on the list.

1,4,9,16,25 \, {\color{red}(30)} \, 36,49…

$30$ lies between two square numbers.

\sqrt{25}=5

\sqrt{36}=6

$\sqrt{30}$ falls between two numbers on the list, which means it is a surd and is an irrational number.

$\sqrt{30}=$ is an irrational number between $5$ and $6.$

Example 2: identifying if the root of an integer is rational or irrational without a calculator

Is $\sqrt[3]{56}$ a rational or an irrational number?

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$56$ is an integer.

Identify what type of root it is and write a list of corresponding powers.

This is a cube root.

The list of cube numbers: $1, 8, 27, 64, 125\dots$

Identify where the integer inside the root falls on the list.

1, 8, 27 \, {\color{red}(56)} \, 64, 125…

$56$ lies between two cube numbers.

$\sqrt[3]{27}=3$

$\sqrt[3]{64}=4$

$\sqrt[3]{56}$ falls between two numbers on the list, which means it is a surd and is an irrational number.

$\sqrt[3]{56}=$ is an irrational number between $3$ and $4.$

Example 3: identifying if the root of a fraction is rational or irrational without a calculator

Is $\sqrt{\cfrac{36}{121}}$ a rational or an irrational number?

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$\cfrac{36}{121}$ is a fraction.

Identify what type of root it is and write a list of corresponding powers.

This is a square root.

The list of square numbers is $1,4,9,16,25,36,49,64,81,100,121,144\dots$

Identify where the integer inside the root falls on the list.

1,4,9,16,25, {\color{red}36}, 49,64,81,100, {\color{red}121}, 144…

Both $36$ and $121$ are square numbers.

$\sqrt{36}=6$

$\sqrt{121}=11$

Therefore, $\sqrt{\cfrac{36}{121}}=\cfrac{\sqrt{36}}{\sqrt{121}}=\cfrac{6}{11} \, .$

$\sqrt{\cfrac{36}{121}}$ is equal to a number on the list, which means that this is not a surd and is therefore a rational number.

$\sqrt{\cfrac{36}{121}}$ is a rational number.

Example 4: identifying if the root of a decimal is rational or irrational without a calculator

Is $\sqrt{2.5}$ a rational or an irrational number?

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$2.5$ is a decimal, so we must convert this to a fraction.

$2.5=\cfrac{25}{10}$

Identify what type of root it is and write a list of corresponding powers.

This is a square root.

The list of square numbers is $1,4,9,16,25,36,49\dots$

Identify where the integer inside the root falls on the list.

1,4,9 \, {\color{red}(10)} \, 16, \, {\color{red}25}, \, 36, 49…

$10$ lies between two square numbers so $\sqrt{10}= \text{ is an irrational number.}$

$25$ is a square number so $\sqrt{25}=5 \text{ which is a rational number.}$

However, both numbers need to be rational for the fraction to be rational.

Because one of the numbers is irrational then the fraction will be irrational.

Example 5: identifying if the root of a decimal is rational or irrational without a calculator

Is $\sqrt[5]{0.00005}$ a rational or an irrational number?

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$0.00005$ is a decimal, so we must convert this to a fraction.

$0.00005=\cfrac{5}{100000}$

Identify what type of root it is and write a list of corresponding powers.

This is a $5$ th root.

The list of integers to the power of $5,$

$1^{5}=1$

$2^{5}=32$

…

$10^{5}=100000$

Identify where the integer inside the root falls on the list.

1, \, {\color{red}(5)}, \, 32, … \, {\color{red}100000}

$5$ lies between $1^{5}$ and $2^{5},$ therefore $\sqrt[5]{5}= \text{ is an irrational number.}$

$100000$ is $10^{5}$ therefore $\sqrt[5]{100000}=10$ which is a rational number.

However, both numbers need to be rational for the fraction to be rational.

Because one of the numbers is irrational then the fraction will be irrational.

Example 6: estimating the value of a surd

Estimate the value of $\sqrt{50}$ to one decimal place without using a calculator.

Check that the number inside the root is either an integer or a fraction; If needed, convert any decimals into fractions.

$50$ is an integer.

Identify what type of root it is and write a list of corresponding powers.

This is a square root.

The list of square numbers is $1,4,9,16,25,36,49,64,\dots$

Identify where the integer inside the root falls on the list.

1,4,9,16,25,36,49, \, {\color{red}(50)}, \, 64,…

$50$ lies between two square numbers.

$\sqrt{49}=7$

$\sqrt{64}=8$

$\sqrt{50}= \text{ is an irrational number between } 7 \text{ and } 8.$

As $50$ is very close to $49,$ then $\sqrt{50}$ will be very close to $7.$ Picturing a number line may also help you to estimate your answer.

Estimate $\sqrt{50} \approx 7.1$

(A calculator gives the answer $7.071067812\dots$ )

Teaching tips for irrational numbers

Students should have a solid understanding of rational numbers before being introduced to irrational numbers.

Provide students with real life examples of irrational numbers, including the diagonal of a unit square or the ratio of the circumference to the diameter of a circle $(\pi).$

While worksheets have their place within the math classroom, consider using interactive technology, like educational apps or online tools, to allow students to explore and manipulate irrational numbers.

Easy mistakes to make

Assuming all fractions are rational numbers
All rational numbers can be written as fractions but not all fractions are rational numbers. If the fraction is in the form $\cfrac{a}{b} \, , \; a$ and $b$ are integers, and $b$ ≠ $0$ , then the number is rational.

However, if $a$ or $b$ are not integers then the fraction could represent an irrational number. For example, $\cfrac{\sqrt{2}}{3}$ is a fraction which is irrational.

Assuming all non-terminating decimals are irrational
All irrational numbers are non-repeating decimals. However not all non-terminating decimals are irrational numbers. Recurring decimals are non-terminating decimals which are rational.

For example, $\sqrt{5}=2.23606…$ is a non terminating decimal which is an irrational number.

$\cfrac{8}{9}=0.888…=0.\dot{8}$ is a repeating decimal; a non terminating decimal which is a rational number.

Writing a negative irrational number incorrectly
Here is an example of a positive irrational number, $\sqrt{3}=1.73205…$

To write the negative irrational number of the same magnitude you must write $-\sqrt{3}$ and not $\sqrt{-3}.$

Try both of these on your calculator.
– $\sqrt{3}=-1.73205…$
$\sqrt{-3}=$ you will get an error message on your calculator.

Not all fractions are rational numbers
The definition of an irrational number is a number that cannot be expressed as a fraction in the form $\cfrac{a}{b}$ , where $a$ and $b$ are integers and $b$ ≠ $0.$ But this does not mean that all fractions are rational.

For example, $\cfrac{\sqrt{2}}{2}$ is a fraction which is also an irrational number because the numerator is irrational. The same would be true if the denominator was irrational.

The square root of a positive number that is not a square number is an irrational number
Square numbers (also known as perfect squares) are the numbers produced by the square of an integer.

$1, 4, 9, 16, 25, 36\dots$

When we take the square root of a square number, the answer is rational; when we take the square root of any other positive integer, the answer is irrational.

For example,

$\sqrt{72}=8.48528…$ is an irrational number because $72$ is NOT a square number.

$\sqrt{12}= 3.46410…$ is an irrational number because $12$ is NOT a square number.

Note that $\sqrt{12}$ can be simplified as $\sqrt{4}\times\sqrt{3}=2\sqrt{3},$ but as $3$ is not a square number then $2\sqrt{3}$ produces an irrational number.

When we take the square root of the quotient of two square numbers, the answer is rational.

For example, $\sqrt{0.09}= \sqrt{\cfrac{9}{100}} = \cfrac{ \sqrt{9}}{ \sqrt{100}} = \cfrac{3}{10} = 0.3.$

Practice irrational numbers questions

\sqrt{64}

\sqrt{16}

\sqrt{4}

\sqrt{84}

\sqrt{64}=8

\sqrt{16}=4

\sqrt{4}=2

\sqrt{84} = \text{ irrational } (\sqrt{81}=9 \text{ and } \sqrt{100}=10 \text{ so } \sqrt{84} \text{ lies between these})

\sqrt[4]{64}

\sqrt{64}

\sqrt[3]{64}

None of these answers are irrational.

\sqrt{64}=8

\sqrt[3]{64}=4

$\sqrt[4]{16}=2$ and $\sqrt[4]{81}=3$ so

$\sqrt[4]{64}=$ is an irrational number between $2$ and $3.$

\sqrt{21}

\cfrac{8}{3}

\sqrt{144}

96 \div 7

\cfrac{8}{3} \text{ rational}

\sqrt{144}=12 \text{ rational}

96 \div 7 = \cfrac{96}{7} = \text{ rational}

\sqrt{21} = \text{ irrational}

$\sqrt{16}=4$ and $\sqrt{25}=5,$ so $\sqrt{21}$ lies between these and is therefore irrational.

\cfrac{7}{2\pi}

\sqrt{6}

\cfrac{\sqrt{5}}{\sqrt{25}}

\cfrac{3\pi}{2\pi}

\cfrac{3\pi}{2\pi}=\cfrac{3}{2} \text{ rational}

Although $\pi$ is an irrational number, the fraction can be simplified to cancel out $\pi$ so this number is rational.

$7$ and $8$

$8$ and $9$

$6$ and $7$

$38$ and $39$

$1,4,9,16,25,36,49,64 \, {\color{red}(76)} \, 81,100,121,$ $144,169$

$76$ lies between $64$ and $81$ on the list of square numbers.

\sqrt{64}=8 \text{ and } \sqrt{81}=9, \text{ so } \sqrt{76} \text{ lies between } 8 \text{ and } 9.

4\pi

\sqrt{145}

11.5

\sqrt{130}

11^2=121, \, 12^2=144

Therefore, the square root of any number between $121$ and $144$ will be an irrational number between $11$ and $12.$

$\sqrt{130}$ is the correct answer from the multiple choice but there are many possible solutions.

For example, $\sqrt{125}$ or $\sqrt{142.576}.$

Irrational numbers FAQs

What is the Euler’s number?

First introduced by Leonhard Euler, the Euler’s number is often noted as an “e” and is a mathematical constant that is equal to approximately $2.71828.$

Who discovered irrational numbers?

The understanding of irrational numbers goes back to ancient Greek mathematicians from $6$ th century BCE. The story is told that Hippasus of Metapontum, a member of the Pythagorean school, was the one who discovered the existence of irrational numbers.

The discovery of irrational numbers was challenged at the time, because it challenged some of the Pythagorean beliefs at the time.

Can you multiply irrational numbers?

When any irrational numbers are multiplied by another nonzero rational number, the product will be an irrational number.

The next lessons are

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