35 Facts About Euclidean 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 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 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 Euclidean algorithm was described only for natural numbers and geometric lengths, but the Euclidean 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.

Related searches

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

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

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

Euclidean algorithm division reduces all the steps between two exchanges into a single step, which is thus more efficient.

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

The extended Euclidean 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 Euclidean algorithm was useful in the Sturm chain method for counting the real roots of polynomials in any given interval.

Euclidean algorithm was the first integer relation 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 algorithm, called The Game of Euclid, which has an optimal strategy.

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

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

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

For if the Euclidean 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 Euclidean algorithm is widely used in practice, especially for small numbers, due to its simplicity.

Binary GCD Euclidean 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 Euclidean algorithm uses the same general principle as the binary Euclidean algorithm to speed up GCD computations in arbitrary bases.

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

Related searches

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

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

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

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

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

Euclidean 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 Euclidean algorithm reduces f inexorably; hence, if can be reduced only a finite number of times, the Euclidean algorithm must stop in a finite number of steps.

Any Euclidean algorithm domain is a unique factorization domain, although the converse is not true.

In 1973, Weinberger proved that a quadratic integer ring with is Euclidean algorithm if, and only if, it is a principal ideal domain, provided that the generalized Riemann hypothesis holds.