A rational number is one that can be written in the form

Note that any fraction can be reduced to the point where the numerator and denominator share no common factors:

So if a number is rational, then it can be written as a fraction of integers with no common factors. This property is essential for a classic (and common IB) proof:

Example

Prove that √2 is irrational.

Suppose that √2 is rational, that is

for a,b∈Z where a and b have no common factors. Squaring both sides:

which means that 2∣a2. But since 2 is prime, that means 2∣a, ie that a=2k for some k∈Z:

which means that 2∣b2. But (again) since 2 is prime, 2∣b.

Therefore 2 divides both a and b. But this contradicts the assumption that a and b have no common factors.

Hence, by contradiction, √2 is irrational.

Notice that this proof does not work, for example, on √4 since 4 is not prime!