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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.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] 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 an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
Given a Gaussian integer z 0, called a modulus, two Gaussian integers z 1,z 2 are congruent modulo z 0, if their difference is a multiple of z 0, that is if there exists a Gaussian integer q such that z 1 − z 2 = qz 0. In other words, two Gaussian integers are congruent modulo z 0, if their difference belongs to the ideal generated by z 0.
Gauss proved [10] that for any prime number p (with the sole exception of p = 3), the product of its primitive roots is congruent to 1 modulo p. He also proved [ 11 ] that for any prime number p , the sum of its primitive roots is congruent to μ ( p − 1) modulo p , where μ is the Möbius function .
Gauss proved [7] [non-primary source needed] that = (,) = {() =,, where p represents an odd prime and a positive integer. That is, the product of the positive integers less than m and relatively prime to m is one less than a multiple of m when m is equal to 4, or a power of an odd prime, or twice a power of an odd prime; otherwise, the product ...
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
Argand called cos φ + i sin φ the direction factor, and = + the modulus; [d] [48] Cauchy (1821) called cos φ + i sin φ the reduced form (l'expression réduite) [49] and apparently introduced the term argument; Gauss used i for , [e] introduced the term complex number for a + bi, [f] and called a 2 + b 2 the norm.
Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). [3] This remained the standard [4] in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉.