| arithmetic in which numbers that are congruent modulo a given number are treated as the same. Compare congruence (def. 2), modulo, modulus (def. 2b). |
| modular arithmetic n. A form of arithmetic dealing with integers in which all numbers having the same remainder when divided by a whole number are considered equivalent: Clocks use modular arithmetic with modulus 12, so 4 hours after 9 o'clock is 1 o'clock. |
modular arithmetic mathematics
(Or "clock arithmetic") A kind of integer arithmetic that reduces all numbers to one of a fixed set [0..N-1] (this would be "modulo N arithmetic") by effectively repeatedly adding or subtracting N (the "modulus") until the result is within this range.
The original mathematical usage considers only __equivalence__ modulo N. The numbers being compared can take any values, what matters is whether they differ by a multiple of N. Computing usage however, considers modulo to be an operator that returns the remainder after integer division of its first argument by its second.
Ordinary "clock arithmetic" is like modular arithmetic except that the range is [1..12] whereas modulo 12 would be [0..11].
(2003-03-28)
modular arithmetic
in its most elementary form, arithmetic done with a count that resets itself to zero every time a certain whole number N greater than one, known as the modulus (mod), has been reached. Examples are a digital clock in the 24-hour system, which resets itself to 0 at midnight (N = 24), and a circular protractor marked in 360 degrees (N = 360). Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of Diophantine equations (particularly those restricted to integer solutions). Generalizations of the subject led to important 19th-century attempts to prove Fermat's last theorem and the development of significant parts of modern algebra.
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