In proving irrationality of these numbers, we will use the result that if a prime p divides a

1) Prove that √2 is an irrational number.

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

⇒ 2b

⇒ 2 | a

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

a = 2c for some integer c.

Squaring both sides

a

2b

⇒ b

⇒ 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.

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2) √5 + √3 is an irrational number.

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)

⇒ 5 = a

5 – 3 = a

2 = a

2a√3/b = a

2a√3/b = (a

2a√3 = (a

√3 = (a

⇒ √3 is a rational number which contradicts our assumption.

So, √5 + √3 is an irrational number.

• 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

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