A fraction is a quotient of two integers — the numerator over the denominator — and despite looking simple, it powers almost every measurement system humans use: cooking (½ cup), construction (5/8" plywood), music (3/4 time), finance (basis points = 1/10000) and probability.

Adding or subtracting fractions requires a common denominator, which is built from the Least Common Multiple of the two denominators. Multiplication and division skip that step — multiplication multiplies tops and bottoms, division multiplies by the reciprocal. Every result is then reduced by dividing both parts by their Greatest Common Divisor (Euclid's algorithm, ~2300 years old and still the fastest).

Some decimals terminate (0.625 = 5/8) and others repeat forever (1/3 = 0.333..., 1/7 = 0.142857142857...). A decimal terminates exactly when its reduced denominator has only 2 and 5 as prime factors. The repeating block length divides the multiplicative order of 10 modulo the denominator — pure number theory in a calculator.

For irrational numbers like π, the calculator's Best-approximation mode finds the closest fraction under a denominator cap. The continued-fraction expansion produces 22/7 (denom < 10), 333/106 (denom < 200), then 355/113 — accurate to 7 digits and famously discovered by Chinese astronomer Zu Chongzhi in the 5th century.

Big-number mode uses JavaScript BigInt so you can compute things like 99999999999999999999/3 exactly without losing a single digit — useful for cryptography, combinatorics and just-for-fun math.

Tip: store an intermediate result with ★ Save, then paste it into the Expression box to chain longer calculations without retyping.