Rational and Irrational Numbers

Fractions and the Numbers Between

The rational numbers $\mathbb{Q}$ are ratios $\frac{a}{b}$ of integers ($b \neq 0$). Every rational has a lowest-terms form, obtained by dividing numerator and denominator by their gcd — the unique representative with $\gcd(a, b) = 1$. Fraction arithmetic follows fixed rules: a common denominator to add ($\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$), and numerator-times-numerator over denominator-times-denominator to multiply.

Rationals have a clean decimal signature: a number is rational exactly when its decimal expansion terminates or eventually repeats. $\frac{1}{4} = 0.25$ (terminates), $\frac{1}{3} = 0.333\dots$ (repeats), $\frac{1}{7} = 0.\overline{142857}$ (repeats with period 6). Conversely, any terminating or repeating decimal can be written as a fraction. This is a complete characterization: terminating-or-repeating $\Leftrightarrow$ rational.

The irrational numbers are the reals that are not rational — their decimals go on forever without repeating. Famous examples are $\sqrt{2}$, $\pi$, and $e$. That $\sqrt{2}$ is irrational is proved by contradiction (Unit 1): assume $\sqrt{2} = \frac{a}{b}$ in lowest terms; then $a^2 = 2b^2$, forcing $a$ even, hence $b$ even too — contradicting lowest terms. So no such fraction exists.

A surprising fact about how these mix: between any two distinct real numbers lie both a rational and an irrational — both sets are dense in the line. Yet the rationals are countable (listable in a sequence) while the irrationals are uncountable, so in a precise sense "almost all" real numbers are irrational, even though the rationals are everywhere.

Common pitfall: believing that any decimal that "looks endless" is irrational. A decimal is irrational only if it is both non-terminating and non-repeating. $0.333\dots$ and $0.\overline{142857}$ run forever but repeat, so they are perfectly rational ($\frac{1}{3}$ and $\frac{1}{7}$). Endlessness alone does not make a number irrational — the absence of a repeating pattern does.