# What are irrational numbers ?

In this chapter we will learn about the concept of irrational numbers with examples.

After reading this post, you will able to identify these numbers on your own.

## Irrational number definition

The numbers that cannot be expressed in the form of p / q are called irrational numbers.

In other words, irrational numbers cannot be written in the form of fraction.

These numbers are non repeating and non terminating decimals whose digits never ends. Due to its non terminating nature, we have to represent it using approximate value.

### Examples of irrational numbers

Given below are some examples of irrational numbers.

(a) The square root of all numbers ( except perfect square ) are irrational numbers.

For example, consider value of \mathtt{\sqrt{2}}

\mathtt{\sqrt{2} =\ 1.41421356\ .\ .\ \ .}

Note that the value of \mathtt{\sqrt{2}} is non repeating and non terminating decimal. You cannot represent the above value in the form of fraction.

Since the decimal value is unending, we take approximate value \mathtt{\sqrt{2}} for calculation purpose.

\mathtt{\sqrt{2} =\ 1.414\ \ ( approx\ value)}

Similarly the value of \mathtt{\sqrt{3} ,\ \sqrt{5} ,\ \sqrt{7} ,\ \sqrt{11} \ } are examples of irrational numbers.

(b) Value of pi ( 𝜋 )

The number 𝜋 is also an irrational number.

The value of 𝜋 is given as;

𝜋 = 3.141592653 . . . . . .

The value of 𝜋 is non terminating and non repeating decimal which cannot be represented in the form of fraction.

For our convenience, the approximate value of 𝜋 is written as;

𝜋 = \mathtt{\frac{22}{7}} or 3.14

The above fraction is just an approximate value, so please don’t get confused that 𝜋 is written in form of fraction.

(c) Euler number (e)

Euler number is extensively used in logarithm chapter ( will be discussed in higher mathematics ).

The value of e is given as;

e = 2.718281828459 . . . . .

Note that the value of e is non repeating and non terminating decimal. Hence, it qualifies as a irrational number.

#### Is \mathtt{\sqrt{9}} an irrational number ?

No !!

Number 9 is a perfect square and square root of perfect square is an integer.

The value of \mathtt{\sqrt{9}} is given as;

\mathtt{\sqrt{9} \ =3}

Since the value is a simple integer, \mathtt{\sqrt{9}} is not an irrational number.

Conclusion:
The square root of perfect square is not an irrational number.

## Difference between rational and irrational number

The rational number can be expressed in the form of fraction p / q, while we cannot do the same with irrational number.

The value of fraction of rational number is terminating or repeating.

Consider the below examples of rational numbers;

(a) 1.42

It’s a terminating decimal, hence a rational number.

(b) 3.9797979797 . . . .

It’s a non terminating but repeating decimal.

Note that the digits 97 are repeating again and again. Such decimals can be easily converted into fraction form, hence they are rational numbers.

(c) 15.584584584 . . . .

Again it is a repeating decimal with digits 584 are repeating again and again. So they come under rational number.

## Important points about rational numbers

(a) The HCF of two irrational number is always 1.

(b) Adding two or more irrational numbers result in irrational number.

(c) Multiplication of two irrational number may or may not result in irrational number.

For example;

\mathtt{\pi \times \pi =\pi ^{2}}

Here \mathtt{\pi ^{2}} is an irrational number.

On the other hand;
\mathtt{\sqrt{3} \times \sqrt{3} =3}

Here 3 is a rational number.

## Significance of irrational numbers

Irrational numbers are important as;

(a) they are used in Pythagoras theorem calculation

(b) irrational number 𝜋 is used in calculation of area of circle, sphere, cylinder etc.

(c) irrational number e ( Euler’s number ) is widely used in logarith calculation.

I hope you understood the basic concept of irrational numbers. Given below are some problems for your practice.

## Identify Irrational number – Solved Problems

(01) Identify if the below numbers are rational or irrational with proper reasoning.

(a) \mathtt{\sqrt{11}}

(b) \mathtt{\frac{6}{13}}

(c) 8.138383838 . . . .

(d) 6,1395307612 . . . .

(e) \mathtt{\sqrt{16}}

Solution

(a) \mathtt{\sqrt{11}}

Square root of all numbers (except perfect square ) are irrational numbers.

(b) \mathtt{\frac{6}{13}}

The number is in form of fraction. Hence it is rational number.

(c) 8.138383838 . . . .

In the decimal, the digits 38 are repeating again and again. Decimals with repeating digits can be converted in fraction form, hence are rational number.

(d) 6.1395307612 . . . .

It’s a non terminating and non repeating decimal. Hence it’s an irrational number.

(e) \mathtt{\sqrt{16}}

16 is a perfect square. Square root of perfect square is a rational number.

\mathtt{\sqrt{16} =4}

Next chapter : Decimal value of irrational number