Rational vs Irrational Numbers

Rational and irrational are the two halves of the real numbers — every real is exactly one or the other.

Rational numbers

A real number is rational if it can be expressed as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$.

Decimal characterisation: rationals have decimals that either terminate ($0.25 = \frac{1}{4}$) or eventually repeat ($0.\overline{3} = \frac{1}{3}$, $0.1\overline{6} = \frac{1}{6}$).

The set of rationals is denoted $\mathbb{Q}$. Despite being dense (between any two rationals there's another rational), the rationals are countable — same cardinality as $\mathbb{N}$.

Irrational numbers

Cannot be expressed as a ratio of integers. Decimals are non-terminating and non-repeating.

Famous irrationals:

• $\pi \approx 3.14159...$
• $e \approx 2.71828...$
• $\sqrt{2} \approx 1.41421...$
• $\phi$ (golden ratio) $= (1 + \sqrt{5})/2$.

The set of irrationals is uncountable — strictly larger than the rationals, even though rationals are dense.

Why this matters

• $\sqrt{2}$ being irrational was a famous Pythagorean discovery (legend: Hippasus was drowned for revealing it).
• $\pi$ being irrational means you can never write it as a fraction.
• The decimal of $1/7 = 0.\overline{142857}$ — the period of repetition is at most $q - 1$.

How to test

If you have a number, ask:

• Decimal terminates → rational.
• Decimal repeats with a clear period → rational.
• Decimal goes on without repetition (e.g. $\pi$, $e$, $\sqrt{2}$) → irrational.

Algebraic tests use closure: rationals are closed under $+, -, \times, /$ (excluding 0). Sum of two irrationals can be rational (e.g. $\sqrt{2} + (-\sqrt{2}) = 0$).