Last modified on August 3rd, 2023

chapter outline


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).

Irrational Numbers

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

Pi  (π)3.1415926535⋅⋅⋅⋅
Euler’s Number (e)2⋅718281⋅⋅⋅⋅
Golden ratio, φ1.6180339⋅⋅⋅⋅

Let us solve an example



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

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’.


  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 × b = irrational number, if 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


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

√7 × √7


√7 × √7
= 7
= rational

Show that √7 + √5 is irrational.


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.

Rational vs Irrational Numbers

Irrational NumbersRational Numbers
Cannot be expressed in the form of a fraction (p/q)Can be expressed in the form of a fraction (p/q)
The decimal expansion is non-terminating (continues endlessly) and non-recurring (with no group of digits repeating) at any pointThe decimal expansion is terminating (ends at some point) or non-terminating recurring (repeating)

Last modified on August 3rd, 2023

3 thoughts on “Irrational Numbers

  1. 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:

    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!!

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