Question

Prove that 2 + 3√5 is an irrational number
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Answer 1

Answer:

so, assume that 2+3root 5 is rational.

2+3root5=p/q,where p and q are co primes and q not equals to 0.

3root 5=p/q-2.

root 5=p/q-2/3.

A rational number never equals to an irrational number.

We assume that p and q are co primes and q not equals to 0.

so, our assumption is wrong.

2+3root 5 is an irrational number.

Step-by-step explanation:

Answer 2

To Prove :

• 2 + 3√5 is an irrational number.

Proof :

Let us assume to the contrary that 2 + 3√5 is a rational number in the form of a/b such that a and b are co primes and b ≠ 0.

Now

→ 2 + 3√5 = a/b

→ 3√5 = a/b - 2

→ 3√5 = a - 2b/ b

→ √5 = a - 2b/3b

Since, a, b, 2 and 3 are integers and a - 2b/3b is rational.

So, √5 is also rational.

But it contradicts the fact that√5 is irrational.

This contradiction has arisen due to our wrong assumption.

Thus, 2 + 3√5 is an irrational number.

Hence, proved !