Proving Irrationality of Numbers

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

In this section, we will discuss proving irrationality of numbers.We will prove √2, √3 and √2/√5 etc. are irrational numbers using Fundamental Theorem of Arithmetic.

In proving irrationality of these numbers, we will use the result that if a prime p divides a2 then it divides ‘a’ also. We will prove the irrationality of numbers by using the method of contradiction.

Examples :
1) Prove that √2 is an irrational number.

Solution :
Let us assume that √2 is a rational number. So,
√2 = a/b (where a and b are prime numbers with HCF = 1)
Squaring both sides
2 = a2/b2
⇒ 2b2 = a2
⇒ 2 | a2
⇒ 2 | a ------------> (1) [by Theorem -> Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer]
a = 2c for some integer c.
Squaring both sides
a2 = 4c2
2b2 = 4c2 [ Since a2=2b2]
⇒ b2 = 2c2
⇒ 2 | b -------------> (2)
∴ From (1) and (2) we obtain that 2 is a common factor of ‘a’ and ‘b’. But this contradicts the fact.
So our assumption of √2 is a rational number is wrong.
Hence, √2 is an irrational number.
---------------------------------------------------------------------------------
2) √5 + √3 is an irrational number.
Solution :
Let √5 + √3 is a rational number equal to a/b.
√5 + √3 = a/b
√5 = a/b - √3
Squaring both sides
5 = (a/b - √3)2
⇒ 5 = a2/b2 - 2a√3/b + 3
5 – 3 = a2/b2 - 2a√3/b
2 = a2/b2 - 2a√3/b
2a√3/b = a2/b2 - 2
2a√3/b = (a2 - 2 b2)/ b2
2a√3 = (a2 - 2 b2)b / b2
√3 = (a2 - 2 b2)/2ab
⇒ √3 is a rational number which contradicts our assumption.
So, √5 + √3 is an irrational number.

Euclid's Geometry

Euclid Geometry
Euclids division lemma
Euclids division Algorithm
Fundamental Theorem of Arithmetic
Finding HCF LCM of positive integers
Proving Irrationality of Numbers
Decimal expansion of Rational numbers

From Euclid Geometry to Real numbers

Home Page

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.