Question

prove that under root 5 is irrational show that 2 + under root 5 is also irrational ​

Answer 1

Answer:

hey mate here is your answer..

let under root 5 is rational no.

under root 5=a/b(a,b is not equal to zero, a and b are co prime ,a and b are integers..)

b under root 5 =a

(b under root 5)^2=(a)^2

5b^2=a^2........(i)

a^2 is divisible by 5

a is divisible by 5

let a=5m

5b^2=a^2

5b^2=25m^2

b^2=5m^2

b^2 is divisible by 5

b is also divisible by 5

which is contradiction to our assumption

so under root 5 is irrational no..

(ii)2+under root 5 is irrational no.

let 2+under root 5 is rational no.

2+under root 5=p/q

under root 5=(p/q)-2

under root 5 =(p-2q)/q

since p and q are integers ,so we get (p-2q)/q is rational

under root 5 is a rational no.

which is contradiction to our assumption

so 2+under root 5 is a irrational no.

hope it may help u

Answer 2

Let us suppose that √5 is rational. Then there exist two positive integers a and B such that

√5 = a/b

Where a and B are co primes

Squaring on both side gives us

5=a^2/b^2

5b^2 = a^2

It means 5 is a factor of a^2 and a as well

5c = a. (as 5 is a factor of a)

Squaring on both sides gives us

25c^2 = a^2

25c^2 = 5b^2. ( As proved above)

b^2 = 5c^2

It means 5 is also a factor of B.

Hence it is a contradiction as a and b were co primes.

Hence our supposition is wrong and √5 is irrational.

For further solution see the pic