Proof that 2√5 is an irrational number

Assumption:
Let's assume that 2√5 is a rational number.

Definition of Rational Number:
A rational number is a number that can be expressed as the ratio of two integers.

Expressing 2√5 as a ratio:
If 2√5 is rational, it can be expressed as a ratio of two integers a and b where b is not equal to 0.
2√5 = a/b

Squaring both sides:
(2√5)^2 = (a/b)^2
4*5 = a^2/b^2
20 = a^2/b^2

Implication:
This implies that a^2 is a multiple of 20.

Understanding the property of squares:
If a number is a perfect square, then all of its prime factors occur in pairs. However, 20 has prime factors 2 and 5, which do not occur in pairs.

Contradiction:
Since 20 does not satisfy the property of perfect squares, it follows that a^2 cannot be a multiple of 20. Hence, our assumption that 2√5 is rational is incorrect.

Conclusion:
Therefore, 2√5 is an irrational number as it cannot be expressed as the ratio of two integers.