Rational and Irrational Numbers

Rational Numbers and Decimals

The rational numbers $\mathbb{Q}$ are ratios $\frac{a}{b}$ of integers ($b \neq 0$). Every rational has a lowest-terms form where $\gcd(a, b) = 1$.

Fraction arithmetic uses these rules:

Decimal Signature: A number is rational iff its decimal expansion terminates or eventually repeats.

• $\frac{1}{4} = 0.25$ (terminating)
• $\frac{1}{3} = 0.\bar{3}$ (repeating)
• $\frac{1}{7} = 0.\overline{142857}$ (period 6)

Irrationals and Real Density

The irrational numbers are reals that are not rational; their decimals run forever without repeating. Examples: $\sqrt{2}$, $\pi$, $e$.

Proof by Contradiction: Assume $\sqrt{2} = \frac{a}{b}$ in lowest terms. Then $a^2 = 2b^2$, forcing both $a$ and $b$ to be even, which violates lowest terms.

Both sets are dense (both appear between any two reals).

Common Pitfall: Believing endless decimals are irrational. $0.333\dots$ runs forever, but it repeats, making it rational.