# Irrational Numbers

Irrational numbers are real numbers that cannot be written as a simple fraction or ratio. In simple words, the irrational numbers are those numbers those are not rational. Hippasus, a Greek philosopher and a Pythagorean, discovered the first evidence of irrational numbers 5th century BC. However, his theory  was not accepted.

Irrational numbers can’t be written as p/q form (ratio), where the denominator, q is not zero (q ≠ 0).

## Common Examples of Irrational Numbers

Few examples of irrational numbers are given below:

1. π (pi), the ratio of a circle’s circumference to its diameter,  is an irrational number. It has a decimal value of 3.1415926535⋅⋅⋅⋅ which doesn’t stop at any point.
2. √x is irrational for any integer x, where x is not a perfect square.

In a right triangle with a base length of 1 unit, the hypotenuse is √2, which is irrational. (√2 = 1⋅414213⋅⋅⋅⋅)

1. Euler’s number, e = 2⋅718281⋅⋅⋅⋅
2. Golden ratio, φ = 1.6180339⋅⋅⋅⋅

Given below are some popular irrational numbers.

### List of Irrational Numbers

Let us solve an example

${\dfrac{5\sqrt{2+4}}{\sqrt{3}}}$

Solution:

{\begin{aligned}\dfrac{5\sqrt{2+4}}{\sqrt{3}}\\ \simeq \dfrac{5\sqrt{6}}{\sqrt{3}}\end{aligned}}
= ${ 5\sqrt{2} }$

## Symbol of Irrational Numbers

Irrational numbers are universally represented by the alphabet ‘P’. The universal symbols for rational numbers is ‘Q’, real numbers is ‘R’.

## Properties

1. Are real numbers only
2. Decimal expansion is non-terminating (continues endlessly)
3. Addition of a rational and irrational number gives an irrational number as the sum; a + b = irrational number, here a = rational number, b = irrational number
4. Multiplication of a rational and irrational number may give a rational number as the product; a × b = irrational number, here a = rational number, b = irrational number
5. Sum or product of 2 irrational numbers may give an irrational number; √2 × √2 = 2.
6. The Least common factor (LCM) for 2 irrational numbers may or may not exist

Let us solve some examples to understand the concept better.

π × π

Solution:

Let us see if π × π = rational or irrational
π × π
= π2
= irrational, ∵ π is irrational

√7 × √7

Solution:

√7 × √7
= 7
= rational

Show that √7 + √5 is irrational.

Solution:

Let us assume that √7 + √5 is rational
If √7 + √5 is rational,
Then, ${\dfrac{7-5}{\sqrt{7}+\sqrt{5}}=\sqrt{7}-\sqrt{5} }$ , implying √7 – √5 is also rational
(√7 + √5) – (√7 – √5)
= 2√5, (implies 2√5 is rational)
Hence, ${2\sqrt{5}\times \dfrac{1}{2}}$
= √5 is also rational. But this is a contradiction
Hence, we get the proof of irrational numbers by contradiction

## Identifying Irrational Numbers

We already have learnt that irrational numbers are real numbers which cannot be represented in the form of p/q, where p and q are integers, and q ≠ 0, and also can’t be simplified to a closed decimal value.

Taking √5 as an example,

It cannot be represented in the form of a fraction

It cannot be simplified either. Its value is 2.236067⋅⋅⋅⋅ and it is not a closed decimal value.

Thus, √5 is irrational.

## Rational Numbers and Irrational Numbers

Rational numbers are numbers that can be expressed in the form of a fraction (p/q) or ratio.

Given below the differences between rational and irrational numbers in a table.