# Euclidean Algorithm - Other Number Systems

Other Number Systems

As described above, the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers). However, it may be generalized to the real numbers, and to more exotic number systems such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory.

### Other articles related to "other number systems, number, numbers":

Euclidean Algorithm - Other Number Systems - Noncommutative Rings
... of the remainder ρ0 must be strictly smaller than β, and there must be only a finite number of possible sizes for ρ0, so that the algorithm is guaranteed to terminate ... Most of the results for the GCD carry over to noncommutative numbers ... In other words, there are numbers σ and τ such that Γright = σα + τβ The analogous identity for the left GCD is nearly the same Γleft = ασ + βτ Béz ...
39 (number) - In Mathematics
39 is the smallest natural number which has three partitions into three parts which all give the same product when multiplied {25, 8, 6}, {24, 10, 5}, {20, 15, 4} ... The thirteenth Perrin number is 39, which comes after 17, 22, 29 (it is the sum of the first two mentioned) ... factor of 392 + 1 = 1522 is 761, which is obviously more than 39 twice, 39 is a Størmer number ...
496 (number) - In Mathematics
496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such ... As a perfect number, it is tied to the Mersenne prime 31, 25 - 1, with 24 ( 25 - 1 ) yielding 496 ... Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8 ...
38 (number)
... This article discusses the number thirty-eight ... disambiguation) ← 39 ... → 38 ← 39 ... → List of numbers — Integers 90 ... → Cardinal thirty-eight Ordinal 38th (thirty-eighth ...
39 (number) - In Other Fields
... with their first, "'39" does in fact fall in the thirty-ninth position The retired jersey number of former baseball player Roy Campanella The book series "The 39 ... History The number of signers to the United States Constitution, out of 55 members of the Philadelphia Convention delegates The traditional number of times citizens ... Japanese Internet chat slang for "thank you" when written with numbers (3=san 9=kyu) Pier 39 in San Francisco The number of the French department Jura In Afghanistan, the number 39 is considered unlucky ...

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In all systems of theology the devil figures as a male person.... Yes, it is women who keep the church going.
Don Marquis (1878–1937)

In this world, which is so plainly the antechamber of another, there are no happy men. The true division of humanity is between those who live in light and those who live in darkness. Our aim must be to diminish the number of the latter and increase the number of the former. That is why we demand education and knowledge.
Victor Hugo (1802–1885)