Irrational Theorems Online Maths NCERT Solutions Class 10

Irrational Theorems Online Maths NCERT Solutions Class 10

Theorem 1

Let P be a prime number and a be a positive integer. If p divides  then show that p divides a.

Proof: – Let p be a prime number and a be a positive integer such that p divides

We all are aware that every positive integer can be expressed as the product of primes.

Let a = p1 , p2 ……. pn where   , ….. , are primes p1,  p2 ……. pn
a2 = (p1, p2 ……. pn) (p1, p2 ……. pn)

a2 = (p12, p22 ……. pn2)

P divides a2

P is one of  p1, p2 ……. pn

P divides a

Theorem 2

Prove that √2 is irrational.

Proof: – Let √2 be rational and let its simplest form be a/b

Then a an b are integers having no common factors other than 1 and b is not equal 0

√2=a/b

2=a²/b²

2b²=a²

2 divides a

2 is prime and divides a²

Let a=2c for some integer c

Putting a=2c we get

2b²=4c²

b²=2c²

2 divide b²

2 divide b

2 is prime and divides b²

2 divide b

2 is a common factor of a and b

But this contradicts the fact that a and b have no common factor other than 1

The contradiction arises by assuming that √2 is rational

Hence √2 is irrational

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