Arithmetic Functions
| Operation | Input | Output | Algorithm | Complexity |
|---|---|---|---|---|
| Addition | Two n-digit numbers | One n+1-digit number | Schoolbook addition with carry | Θ(n) |
| Subtraction | Two n-digit numbers | One n+1-digit number | Schoolbook subtraction with borrow | Θ(n) |
| Multiplication | Two n-digit numbers |
One 2n-digit number | Schoolbook long multiplication | O(n2) |
| Karatsuba algorithm | O(n1.585) | |||
| 3-way Toom–Cook multiplication | O(n1.465) | |||
| k-way Toom–Cook multiplication | O(nlog (2k − 1)/log k) | |||
| Mixed-level Toom–Cook (Knuth 4.3.3-T) | O(n 2√2 log n log n) | |||
| Schönhage–Strassen algorithm | O(n log n log log n) | |||
| Fürer's algorithm | O(n log n 2log* n) | |||
| Division | Two n-digit numbers | One n-digit number | Schoolbook long division | O(n2) |
| Newton–Raphson division | O(M(n)) | |||
| Square root | One n-digit number | One n-digit number | Newton's method | O(M(n)) |
| Modular exponentiation | Two n-digit numbers and a k-bit exponent | One n-digit number | Repeated multiplication and reduction | O(2kM(n)) |
| Exponentiation by squaring | O(k M(n)) | |||
| Exponentiation with Montgomery reduction | O(k M(n)) |
Schnorr and Stumpf conjectured that no fastest algorithm for multiplication exists.
Read more about this topic: Computational Complexity Of Mathematical Operations
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“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)