Arithmetic Operations

Four moves — add, subtract, multiply, divide — that compose into every later calculation.

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The brief

Add, subtract, multiply, divide. Four moves taught to every child in every educated society on Earth, learned so completely that adults forget they were ever difficult. Each move, however, represents a real conceptual leap: subtraction extends counting backward; multiplication compresses repeated addition; division inverts multiplication and is messy when the answer isn't a whole number. The Babylonians had multiplication tables on clay tablets by 1800 BCE. The Egyptians multiplied by doubling and adding. The Indians invented the place-value algorithms — long multiplication, long division — that an educated person can still execute on paper today. The four operations are arithmetic; everything else in mathematics is built on them.

Addition combines counts: 3 + 5 = 8. It is commutative (a + b = b + a), associative ((a + b) + c = a + (b + c)), and has identity 0. Subtraction is the inverse: a − b = c means a = b + c. Subtraction is not commutative or associative. Multiplication is repeated addition: 3 × 4 = 4 + 4 + 4 = 12. It is commutative, associative, has identity 1, and distributes over addition: a × (b + c) = a × b + a × c — the distributive law, the bridge between addition and multiplication, on which all subsequent algebra rests. Division is the inverse of multiplication: a ÷ b = c means a = b × c. Division by zero is undefined (no c satisfies the equation when b = 0 and a ≠ 0). Order of operations — parentheses, exponents, multiplication and division (left-to-right), addition and subtraction (left-to-right), often abbreviated PEMDAS — is a notational convention preventing ambiguity. Algorithms for the four operations come in many flavors: the standard long-multiplication and long-division algorithms are fast and learnable; abacus-based mental arithmetic persists in much of East Asia; Karatsuba and Schönhage-Strassen are the subquadratic algorithms big-integer libraries use to multiply numbers with millions of digits; Harvey-van der Hoeven reaches the theoretical optimum, but only for numbers far larger than any that arise.

Why nowMost adults now outsource arithmetic to calculators, phones, or spreadsheets, and mental arithmetic gets far less daily practice than it did a generation ago. Underneath, however, the four operations run every numerical computation in every electronic device on the planet, performed billions of times per second per CPU core. Floating-point arithmetic — the standard for most engineering and scientific work — is a precise approximation of the four operations on (most of) the real numbers, with well-defined rules about rounding error. Big-integer arithmetic, in libraries like GMP, supports cryptographic computations on integers of thousands of digits. Modular arithmetic, polynomial arithmetic, matrix arithmetic — every later kind of arithmetic — is built on the same four moves applied to richer objects.