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In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae , published in 1801.
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. [1] The concept of modularity is used primarily to reduce complexity by breaking a system into varying degrees of interdependence and independence across and "hide the complexity of each part behind an abstraction and interface". [2]
Using a residue numeral system for arithmetic operations is also called multi-modular arithmetic. Multi-modular arithmetic is widely used for computation with large integers, typically in linear algebra, because it provides faster computation than with the usual numeral systems, even when the time for converting between numeral systems is taken ...
In mathematics, a modular form is a (complex) analytic function on the upper half-plane, , that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with ...
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo n {\displaystyle n} is a congruence relation on the ring of integers, and arithmetic modulo n {\displaystyle n} occurs on the corresponding quotient ring .
In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called ' clock arithmetic ', where numbers 'wrap around' after they reach a certain value (the modulus ). For example, when the modulus is 12, then any two numbers that leave the same ...
Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [1] In the standard notation of modular arithmetic this congruence is written as.
Modular equation. In mathematics, a modular equation is an algebraic equation satisfied by moduli, [1] in the sense of moduli problems. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli. The most frequent use of the term modular equation is in ...