Introduction:
To prove that root 2 is irrational, we need to show that it cannot be expressed as a ratio of two integers. Otherwise, if it can be expressed as a ratio of two integers, it would be a rational number.

Assumption:
Assuming that root 2 can be expressed as a ratio of two integers, we can simplify the ratio to the lowest possible terms, i.e., the greatest common divisor of the numerator and denominator is 1.

Derivation:
Let us assume that root 2 can be expressed as a ratio of two integers, say a and b, where a and b are co-prime. Then, we can write:

root 2 = a/b

Squaring both sides, we get:

2 = a^2 / b^2

Multiplying both sides by b^2, we get:

2b^2 = a^2

Therefore, a^2 is even, which implies that a is also even. Let a = 2c, where c is an integer. Substituting this value in the above equation, we get:

2b^2 = (2c)^2

Simplifying this, we get:

b^2 = 2c^2

Therefore, b^2 is even, which implies that b is also even. But this contradicts our assumption that a and b are co-prime, because both a and b have a common factor of 2. Hence, our assumption that root 2 can be expressed as a ratio of two integers is false.

Conclusion:
Therefore, we can conclude that root 2 is an irrational number, which cannot be expressed as a ratio of two integers.