V11Prove that V2 is an irrational number. Hence, show that 3 - V2 is a

1.	V11Prove that V2 is an irrational number. Hence, show that 3 - V2 is an irrationanumber.
Answer» Step-by-step explanation: ANSWER: 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 i hope this is helpful for you so please FOLLOW me and mark me as brainlists

V11Prove that V2 is an irrational number. Hence, show that 3 - V2 is an irrationanumber.

Answer»

Step-by-step explanation:

ANSWER:

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

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