32 Facts About Euclid's algorithm

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor of two integers, the largest number that divides them both without a remainder.

Euclidean Euclid's algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number.

Version of the Euclidean Euclid's algorithm described above can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other.

The original Euclid's algorithm was described only for natural numbers and geometric lengths, but the Euclid's algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable.

Euclidean algorithm calculates the greatest common divisor of two natural numbers a and b The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder.

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Since the last remainder is zero, the Euclid's algorithm ends with 21 as the greatest common divisor of 1071 and 462.

Euclidean Euclid's algorithm can be visualized in terms of the tiling analogy given above for the greatest common divisor.

At every step k, the Euclidean Euclid's algorithm computes a quotient qk and remainder rk from two numbers rk-1 and rk-2.

Centuries later, Euclid's algorithm was discovered independently both in India and in China, primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.

The Euclidean Euclid's algorithm was first described numerically and popularized in Europe in the second edition of Bachet's Problemes plaisants et delectables .

The extended Euclidean Euclid's algorithm was published by the English mathematician Nicholas Saunderson, who attributed it to Roger Cotes as a method for computing continued fractions efficiently.

In 1829, Charles Sturm showed that the Euclid's algorithm was useful in the Sturm chain method for counting the real roots of polynomials in any given interval.

Euclidean Euclid's algorithm was the first integer relation Euclid's algorithm, which is a method for finding integer relations between commensurate real numbers.

In 1969, Cole and Davie developed a two-player game based on the Euclidean Euclid's algorithm, called The Game of Euclid, which has an optimal strategy.

The Euclidean Euclid's algorithm has other applications in error-correcting codes; for example, it can be used as an alternative to the Berlekamp–Massey Euclid's algorithm for decoding BCH and Reed–Solomon codes, which are based on Galois fields.

Euclid's algorithm can be used to solve multiple linear Diophantine equations.

Euclidean Euclid's algorithm can be used to arrange the set of all positive rational numbers into an infinite binary search tree, called the Stern–Brocot tree.

Euclidean Euclid's algorithm has almost the same relationship to another binary tree on the rational numbers called the Calkin–Wilf tree.

For if the Euclid's algorithm requires N steps, then b is greater than or equal to FN+1 which in turn is greater than or equal to f, where f is the golden ratio.

Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity.

Binary GCD Euclid's algorithm is an efficient alternative that substitutes division with faster operations by exploiting the binary representation used by computers.

Additional efficiency can be gleaned by examining only the leading digits of the two numbers a and b The binary algorithm can be extended to other bases, with up to fivefold increases in speed.

Lehmer's GCD Euclid's algorithm uses the same general principle as the binary Euclid's algorithm to speed up GCD computations in arbitrary bases.

Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements.

The goal of the Euclid's algorithm is to identify a real number such that two given real numbers, and, are integer multiples of it: and, where and are integers.

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Second, the Euclid's algorithm is not guaranteed to end in a finite number of steps.

The Euclid's algorithm is unlikely to stop, since almost all ratios of two real numbers are irrational.

The Euclidean Euclid's algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can be defined.

Polynomial Euclidean Euclid's algorithm has other applications, such as Sturm chains, a method for counting the zeros of a polynomial that lie inside a given real interval.

In general, the Euclidean Euclid's algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments.

Euclidean Euclid's algorithm developed for two Gaussian integers and is nearly the same as that for ordinary integers, but differs in two respects.

The basic principle is that each step of the Euclid's algorithm reduces f inexorably; hence, if can be reduced only a finite number of times, the Euclid's algorithm must stop in a finite number of steps.