Question

prove that root 5 is an irrational number hence show that 3 + 5 root under is also an irrational number

Answer 1

I know but not clear ly

Answer 2

Let us assume that √5 is irrational.

So it can be expressed in the form p/q where p and q are co prime numbers.

√5=p/q

Squaring both sides

5=p^2/q^2

5q^2=p^2---------(1)

Which means that 5 is a factor of p^2

So 5 is a factor of p

So let p=5m for any integer m

Squaring both sides

P^2=25m^2

Using (1)

5q^2=25m^2

q^2=5m^2

which means that 5 is a factor of q^2

So 5 is a factor of q

We have shown that p and q both have 5 as a common factor but this contradicts the fact that p and q are co prime

So our assumption was wrong

√5 is irrational ------(2)

Now we have to prove that 3+√5 is irrattional .

So let us assume that 3+√5 is rational.

So it can be expressed in the form p/q where p and q are co prime integers.

3+√5=p/q

√5=p/q-3

√5=(p-3q)/q ---- taking q as LCM

Here p,3,q are integers

So (p-3q)/3 is rational . So RHS is rational which means that LHS is rational which means that √5 is rational but we have proved that √5 is irrational {in (2)}

Hence our assumption was wrong

3+√5 is irrational