Every culture invented some way to handle parts of a whole. The Egyptians used unit fractions (1/n), expressing 2/5 as 1/3 + 1/15 because the notation insisted on numerator 1. The Babylonians used base-60 fractions, which is why we still have 60-minute hours and 360-degree circles. The Greeks spoke of ratios between commensurable quantities. The unifying object — a number that is the quotient of two integers — was named rational (from ratio) and became one of the foundations of arithmetic. Until the Pythagoreans found a length that no rational number could express, ratios were assumed to be enough.
A rational number is a number expressible as a fraction p/q where p and q are integers and q ≠ 0. The set is denoted ℚ. Two fractions are equal if their cross-multiplication agrees: p/q = r/s iff ps = qr. The equivalence classes under this relation are the actual rational numbers; 1/2 and 2/4 and 50/100 are the same rational. Operations on rationals: addition by common-denominator (p/q + r/s = (ps + qr)/qs); multiplication straightforward (p/q × r/s = pr/qs); division (p/q ÷ r/s = ps/qr, when r ≠ 0). The four operations make ℚ a field — every nonzero element has a multiplicative inverse — and ℚ is the smallest field containing the integers. Decimal expansions: a rational has a terminating expansion (e.g., 1/4 = 0.25) iff its denominator in lowest terms has only 2 and 5 as prime factors; otherwise the expansion is eventually periodic (1/3 = 0.333…, 1/7 = 0.142857142857…). Every periodic decimal is rational. Density: between any two distinct rationals, there is another rational (and therefore infinitely many) — ℚ has no gaps in this sense. Cardinality: despite their density, the rationals are countable — Cantor's pairing function lists them all. The rationals, in the precise sense of Cantor, are not bigger than the integers.
Most everyday numerical computation uses rationals (or, more often, finite-precision approximations to them). Computer scientists distinguish exact rational arithmetic — slow, big-integer-based, used in computer algebra systems and certified numerical software — from floating-point — fast, approximate, used everywhere performance matters. Continued fractions (a representation of rationals and irrationals as nested integer divisions) appear in the theory of best rational approximation — the basis of musical scales, gear ratios, and the Stern-Brocot tree used in some animation algorithms. Probability is fundamentally rational arithmetic when sample spaces are discrete (the probability of rolling a six is 1/6, an exact rational). Music theory: just intonation uses small-integer rational ratios for harmonious intervals.