# 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 â‰  0).

## Common Examples of Irrational Numbers

Few examples of irrational numbers are given below:

1. Ï€ (pi), the ratio of a circle’s circumference to its diameter,  is an irrational number. It has a decimal value of 3.1415926535â‹…â‹…â‹…â‹… which doesn’t stop at any point.
2. âˆšx 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 âˆš2, which is irrational. (âˆš2 = 1â‹…414213â‹…â‹…â‹…â‹…)

1. Euler’s number, e = 2â‹…718281â‹…â‹…â‹…â‹…
2. Golden ratio, Ï† = 1.6180339â‹…â‹…â‹…â‹…

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 will always give an irrational number; here, a Ã— b = irrational number, if a = rational number, b = irrational number
5. Sum or product of 2 irrational numbers may give an irrational number; âˆš2 Ã— âˆš2 = 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.

Ï€ Ã— Ï€

Solution:

Let us see if Ï€ Ã— Ï€ = rational or irrational
Ï€ Ã— Ï€
= Ï€2
= irrational, âˆµ Ï€ is irrational

âˆš7 Ã— âˆš7

Solution:

âˆš7 Ã— âˆš7
= 7
= rational

Show that âˆš7 + âˆš5 is irrational.

Solution:

Let us assume that âˆš7 + âˆš5 is rational
If âˆš7 + âˆš5 is rational,
Then, ${\dfrac{7-5}{\sqrt{7}+\sqrt{5}}=\sqrt{7}-\sqrt{5} }$ , implying âˆš7 – âˆš5 is also rational
(âˆš7 + âˆš5) – (âˆš7 – âˆš5)
= 2âˆš5, (implies 2âˆš5 is rational)
Hence, ${2\sqrt{5}\times \dfrac{1}{2}}$
= âˆš5 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 â‰  0, and also can’t be simplified to a closed decimal value.

Taking âˆš5 as an example,

It cannot be represented in the form of a fraction

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

Thus, âˆš5 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.

## 3 thoughts on “Irrational Numbers”

1. saron antemeh says:

it is really good even the interface and every thing is just amazing

2. Hamza Siddiqui says:

Hey there! Wonderful little article, it explains things pretty clearly and introduces concepts that seem worth looking into, particularly the properties of irrational numbers. However, I find a lot of confusion with property #4, it seems to be incorrect. See this simple proof by contradiction: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:irrational-numbers/x2f8bb11595b61c86:sums-and-products-of-rational-and-irrational-numbers/v/proof-that-rational-times-irrational-is-irrational

There isn’t any example showcasing this property either in the examples section either, so I’m just assuming it was written in error. Would appreciate a response clearing up the confusion or a correction of the error if it is one, thank you!!

1. staff says:

Thank you for your comment. We have edited that section.