The Real Number System

Numbers sort into nested groups.

Counting numbers: 1, 2, 3, … Add zero and you get whole numbers. Add negatives and you get integers.

Rational numbers are anything writable as a fraction of two integers: 3/4, −5 (which is −5/1), 0.25, and 0.333… since that is 1/3. As decimals they either stop or repeat.

Irrational numbers cannot be written as such a fraction. Their decimals run forever without repeating. √2, π and e are the famous ones.

Together, rationals and irrationals make the real numbers — everything on the number line.

The containments are strict: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ. Each extension exists to make an operation total. ℤ makes subtraction always possible, ℚ makes division by nonzero always possible, and ℝ fills the gaps ℚ leaves — the completeness property, which guarantees that a length like the diagonal of a unit square corresponds to an actual number.

A real number is rational exactly when its decimal expansion terminates or is eventually periodic. √2 is irrational by the classical proof: assuming √2 = p/q in lowest terms forces both p and q to be even, contradicting the assumption.

For classification questions, note that a radical is not automatically irrational — √16 = 4 is an integer — and that operations do not preserve irrationality: √2 · √2 = 2 and π + (−π) = 0 are both rational. The rationals are closed under the four operations (excluding division by zero); the irrationals are closed under none of them.