Let us assume that 2+3√5 is a rational number in the form of p/q.

so that,

2+3√5=p/q
3√5=p/q-2
3√5=p-2q/q
√5=p-2q/3q

here p,q,3 and 2 are integers which are equal to √5, which means √5 is also a rational number.

But this contradict the fact that √5 is an irrational number.

So, our assumption is a contradiction, 2+3√5 is a irrational number.


Thank you.. :-)

Proof that √5 is irrational:
To prove that √5 is irrational, we can use the method of contradiction.

Assumption:
Let's assume that √5 is rational, which means it can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0. We will assume that p/q is in its simplest form, i.e., p and q have no common factors.

Squaring both sides:
If √5 = p/q, then squaring both sides of the equation gives us 5 = (p^2)/(q^2).

Derivation:
From the equation 5 = (p^2)/(q^2), we can derive that p^2 = 5q^2. This implies that p^2 is divisible by 5.

Case 1:
If p is divisible by 5, then p^2 is also divisible by 5.

Case 2:
If p is not divisible by 5, then p^2 is not divisible by 5.

Therefore, we can conclude that p^2 is divisible by 5 for any value of p.

Consequence:
From the above derivation, we can conclude that p^2 is divisible by 5, which means p is also divisible by 5.

Substitution:
Let's substitute p = 5k, where k is an integer, into the equation p/q = √5.

This gives us (5k)/q = √5.

Squaring both sides:
Squaring both sides of the equation (5k)/q = √5 gives us (25k^2)/(q^2) = 5.

Simplifying the equation further, we get 5k^2 = q^2.

Derivation:
From the equation 5k^2 = q^2, we can derive that q^2 is divisible by 5.

Case 1:
If q is divisible by 5, then q^2 is also divisible by 5.

Case 2:
If q is not divisible by 5, then q^2 is not divisible by 5.

Therefore, we can conclude that q^2 is divisible by 5 for any value of q.

Contradiction:
From the above derivation, we can conclude that q^2 is divisible by 5, which means q is also divisible by 5.

This contradicts our initial assumption that p and q have no common factors.

Conclusion:
Since our initial assumption leads to a contradiction, we can conclude that √5 is irrational.

Implication:
As a result, if √5 is irrational, then 2√5 is also irrational since multiplying an irrational number by a rational number does not change its irrationality. Therefore, two times the square root of five is an irrational number.