Question

1. prove that 1/2 - √5 is irrational no.

Answer 1

Let 1/2-√5 is a rational number.
A rational number can be written in the form of p/q where p,q are integers.

1/2-√5=p/q
1/2-p/q=√5
√5=(q-2p)/2q

If p,q are integers then (q-2p)/2q is a rational number.
Then,√5 is also a rational number.
But this contradicts the fact that √5 is an irrational number.
So,our supposition is false.
Therefore,1/2-√5 is an irrational number

Hence proved.

Answer 2

Answer:

Let 1/2-root5 be a rational no.

1/2-root5=p/q

q=p(2-root5)

q/p=2-root5

q/p -2 = -root5

(q-2p)/p = -root5

-(q-2p)/p= root5

p and q are integer the -(q-2p)/p is rational no.

Then, root5 is also a rational no.

But this contradicts the fact that root5 is an irrational number

So,our assumption is wrong

Therefore 1/2-root5 is an irrational number.

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