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 c**annot 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 r**ational 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**