# Proving Irrationality of Numbers

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 a

^{2}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 = a

^{2}/b

^{2}

⇒ 2b

^{2}= a

^{2}

⇒ 2 | a

^{2}

⇒ 2 | a ------------> (1) [by Theorem -> Let p be a prime number. If p divides a

^{2}, then p divides a, where a is a positive integer]

a = 2c for some integer c.

Squaring both sides

a

^{2}= 4c

^{2}

2b

^{2}= 4c

^{2}[ Since a

^{2}=2b

^{2}]

⇒ b

^{2}= 2c

^{2}

⇒ 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 = a

^{2}/b

^{2}- 2a√3/b + 3

5 – 3 = a

^{2}/b

^{2}- 2a√3/b

2 = a

^{2}/b

^{2}- 2a√3/b

2a√3/b = a

^{2}/b

^{2}- 2

2a√3/b = (a

^{2}- 2 b

^{2})/ b

^{2}

2a√3 = (a

^{2}- 2 b

^{2})b / b

^{2}

√3 = (a

^{2}- 2 b

^{2})/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

• 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**

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