The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means "a small measure. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. In Linguistics, ablative case ( abbreviated ABL) is a name given to cases in various languages whose common characteristic " It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Disquisitiones Arithmeticae is a textbook of Number theory written by German Mathematician Carl Friedrich Gauss in 1798 Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Ever since, however, "modulo" has gained many meanings, some exact and some imprecise.

• (This usage is from Gauss's book. ) Given the integers a, b and n, the expression a ≡ b (mod n) (pronounced "a is congruent to b modulo n") means that a − b is a multiple of n. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French For more details, see modular arithmetic. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

• In computing, given two numbers (either integer or real), a and n, a modulo n is the remainder after numerical division of a by n, under certain constraints. Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. See modulo operation.

• Two members of a ring or an algebra are congruent modulo an ideal if the difference between them is in the ideal. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

• Two members a and b of a group are congruent modulo a normal subgroup iff ab⁻¹ is a member of the normal subgroup. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. ↔ See quotient group and isomorphism theorem. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

• Two subsets of an infinite set are equal modulo finite sets precisely if their symmetric difference is finite, that is, you can remove a finite piece from the first infinite set, then add a finite piece to it, and get as result the second infinite set. In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets but not in both

• The most general precise definition is simply in terms of an equivalence relation R. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" We say that a is equivalent or congruent to b modulo R if aRb.

• In the mathematical community, the word modulo is also used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C". See modulo (jargon). This article discusses the usage of the term Modulo as a form of Mathematical jargon.

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