Question

Prove that√2 is an irrational number.Hence,show that 3-√2 is an irrational number

Answer 1

Answer:

Step-by-step explanation:

let us assume that √2 is a rational number

∴we find co prime a and b(≠0) such that √2=a/b

⇒b√2=a

squaring on both sides and rearranging the term, we get

2b²=a²

∴2 divides a² and

 2 divides a also (∵theorem)

let a=2c for some integer c

∴2b²=4c²

⇒b²=2c²

∴2 divides b² and

2 divides b also (∵theorem)

∴a and b have atleast 2 as their common factor

but this contradicts the fact that a and b are co prime .

this contradiction has arisen because of our incorrect assumption that √2 is rational

so we can conclude that √2 is irrational .

(ii)let us assume that 3/√2 is rational

then we find co prime a and b(≠0) such that 3/√2=a/b

∴3b=√2a

⇒3b/a=√2

∴√2 is rational (∵ a and b are integers and ∴3b/a is also rational)

but this contradicts the fact that √2 is irrational

so we can conclude that 3/√2 is irrational