arbitraryprecision

Arbitrary-precision arithmetic refers to computational methods for numbers whose precision is limited only by available memory, rather than by a fixed bit width. It enables exact representation and manipulation of integers, and often of rational or decimal values, without rounding errors inherent to fixed-precision types. This capability is essential when numbers can grow beyond standard data type limits or when exact results are required.

Most implementations represent numbers as sign-m magnitude plus a sequence of digits or limbs in a chosen base (commonly 2^32 or 2^64), stored in a fixed order such as little-endian. Arithmetic is performed with multi-precision algorithms: addition and subtraction extend schoolbook methods, multiplication uses Karatsuba, Toom-Cook, or Schönhage-Strassen approaches for large operands, and division relies on long-division or Newton-Raphson-based methods. Modular arithmetic for cryptography employs efficient reduction techniques, such as Montgomery reduction. For very large numbers, fast Fourier transform (FFT)-based multiplication often dominates performance.

Applications span cryptography, computational number theory, exact scientific calculations, and symbolic computation. Arbitrary-precision decimals are used when exact scaling and precision control are required in financial software. The primary trade-offs are speed and memory usage, which grow with operand size and the sophistication of the chosen algorithms.

Prominent implementations and libraries include GMP (GNU Multiple Precision Arithmetic Library) and MPIR; language-embedded facilities such as Python's int type, Java's BigInteger, .NET's BigInteger, and JavaScript's BigInt. Many systems provide both integer and rational support, and some offer exact decimal arithmetic via dedicated decimal types.