arbitrary-precision arithmetic
Hardware integers live in a fixed-size box — 64 bits, holding numbers up to about 9.2 quintillion — and overflow if you exceed it. But mathematics has no such ceiling: 100! has 158 digits, and a cryptographic key is hundreds of digits long. Arbitrary-precision arithmetic is the trick of representing a number across as many machine words as it needs, so it can grow to thousands or millions of digits, limited only by memory, and every operation on it is exact.
The idea is grade-school arithmetic in a bigger base. A bignum is stored as an array of 'digits', each digit really a full machine word (so the base is 2^32 or 2^64, not 10). Addition carries from one word to the next just as you carry from one column to the next on paper; multiplication of two n-word numbers the schoolbook way costs O(n^2) word-multiplications, and clever algorithms (Karatsuba, then FFT-based methods) bring huge numbers down to nearly O(n log n). Because nothing is ever rounded, 2^1000 is computed to its exact last digit and large factorials, exact powers, and big determinants come out perfectly. This is the bedrock under a computer-algebra system: every exact integer and the numerator and denominator of every exact fraction is a bignum.
Arbitrary precision matters wherever exactness is non-negotiable: cryptography (RSA multiplies numbers hundreds of digits long), number theory, and the exact integer and rational arithmetic inside symbolic computation. The honest caveat is cost — bignums are far slower than the single hardware instruction a 64-bit add takes, and their size (and the time to operate on them) grows with the numbers themselves. This is one engine of expression swell: an exact computation can be correct yet crawl, because its intermediate numbers have quietly ballooned to thousands of digits.
100! (one hundred factorial) is 9332621544...0000000000, a 158-digit integer with twenty-four trailing zeros, computed exactly by a bignum library. A 64-bit machine integer overflows already at 21! (about 5.1 * 10^19), silently wrapping to a wrong, smaller number — the very failure arbitrary precision exists to prevent.
Exact 158-digit factorial: limited by memory, not by a fixed word size.
Arbitrary precision is not the same as floating-point with more digits. A bignum integer or fraction is exact; an arbitrary-precision FLOAT still rounds, just at a chosen higher precision. Exactness comes from integers and rationals, not from carrying more decimal places.