For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a " modulus " ( division ) instruction available rather than just subtraction ( or worse: just Minsky's " decrement ").

This algorithm is much less prone to loss of precision due to massive cancellation, but might not be as efficient because of the division operation inside the loop.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q.

One such algorithm is long division, taught to many school children.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

Thus the division form of Euclid's algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the remainder obtained by dividing the larger number by the smaller number.

A much more efficient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

At this stage in the algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or division by zero is not an issue, despite this implication in the factored equation:

By hand as well as with a computer, this division can be computed by the polynomial long division algorithm.

In dynamic time division multiple access, a scheduling algorithm dynamically reserves a variable number of time slots in each frame to variable bit-rate data streams, based on the traffic demand of each data stream.

Modern computers compute division by methods that are faster than long division: see Division algorithm.

Those against the Wild Card see it as diminishing the importance of the pennant race and the regular season, with the true race often being for second rather than first place, while those in favor of it view it as an opportunity for teams to have a shot at the playoffs even when they have no chance of a first-place finish in their division, thus maintaining fan interest later in the season.

(: see ) In effect, the 1662 Prayer Book marked the end of a period of just over 100 years, when a common form of liturgy served for almost all Reformed public worship in England ; and the start of the continuing division between Anglicans and Nonconformists.

* Coca Cola Championship, the second division of football in England, see Football League Championship

Here we see the clear division of male-female attributes which confined the sexes to specific roles, under Rousseau's popular doctrines.

According to Kallistos Ware, some Western theologians, both Roman Catholic and Anglican, see the theology of Palamas as introducing an inadmissible division within God ; however, others have incorporated his theology into their own thinking, maintaining, as Jeffrey D. Finch reports, that there is no conflict between his teaching and Roman Catholic thought.

It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide.

This includes those who are critical of the division between the two and see them as two aspects of the same movement, and believe that late modernism continues.

For the history of Korea before its division, see History of Korea.

Influential to thinkers associated with Postmodernism are Heidegger's critique of the subject-object or sense-knowledge division implicit in Rationalism, Empiricism and Methodological Naturalism, his repudiation of the idea that facts exist outside or separately from the process of thinking and speaking them ( however, Heidegger is not specifically a Nominalist ), his related admission that the possibilities of philosophical and scientific discourse are wrapped up in the practices and expectations of a society and that concepts and fundamental constructs are the expression of a lived, historical exercise rather than simple derivations of external, apriori conditions independent from historical mind and changing experience ( see Johann Gottlieb Fichte, Heinrich von Kleist, Weltanschauung and Social Constructionism ), and his Instrumentalist and Negativist notion that Being ( and, by extension, reality ) is an action, method, tendency, possibility and question rather than a discreet, positive, identifiable state, answer or entity ( see also Process Philosophy, Dynamism, Instrumentalism, Pragmatism and Vitalism ).

The primary division was between the French and the Imperials ( the followers of the Holy Roman Emperor ), the Imperials wishing to see the Council of Trent reconvened, the French wishing to see it dropped ; a further bone of contention was the Duchy of Parma, for which both the Emperor and the Farnese had claimants.

On the other hand, people tend to specialize just to be able to form groups with high synergy ( see also division of labor and teamwork ).

: For " administrative division " in the sense of a company department, see: Division ( business ).

To ensure self-renewal, stem cells undergo two types of cell division ( see Stem cell division and differentiation diagram ).

In terms of regional jurisdiction there was no precise division between the four Tetrarchs, and this period did not see the Roman state actually split up into four distinct sub-empires.

The circumstances of the 2 artists ' collaboration are unclear ; since Masolino was considerably older, it seems likely that he brought Masaccio under his wing, but the division of hands in the Virgin and Child with Saint Anne is so marked-Masolino is believed to have painted the figure of St. Anne and the angels that hold the cloth of honor behind her, while Masaccio painted the more important Virgin and Child on their throne-that it is hard to see the older artist as the controlling figure in this commission.

Voivodeships are further divided into powiats ( counties ) and gminas ( communes or municipalities ): see Administrative division of Poland.

1 and − 1 divide ( are divisors of ) every integer, every integer ( and its negation ) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself ( see also division by zero ).

Please see the list of the yūshō winners since 1909 for the top division and second division.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.

The Euclidean division that generalizes the Euclidean division of the integers.

An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials ( like Euclidean division ) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for X.

The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F is a Euclidean domain.

but for this discussion the most important division is between those who have been reconstructed and those who haven't.

The liberal-conservative division, we might observe in passing, is not of itself directly involved in a private interest conflict nor even in struggle between ruling groups.

The division is not between those who wish to preserve what they have and those who want change.

Rather it is a division established by two absolutely different ways of thought with regard to man's life in society.

Thus there is a clearer division of authority, administrative and legislative.

A recent study on radiation exposure by the AEC's division of biology and medicine stated: `` The question of the biological effect of ( radiation ) doses is not considered '' herein.

( Note: So far as State Police cars are concerned, only their replacement is under this division ).

The symbol for division is a straight line that separates two numbers placed one above the other.

It is hypothesized that fertility is a function of the social system when the population as a whole is considered and a function of the subsystems when the two-fold division of core families and marginal families is considered.

The simplest division, and the one most frequently used ( with subdivisions ) in gas and electric rate cases, is a threefold division of the total operating and capital costs into `` customer costs '', `` energy '' or `` volumetric costs '', and `` demand '' or `` capacity '' costs.

for this division is but an abstract representation of the social struggle between mysticism and science.

their passions were genuine, and the division between them and the abolitionists is clear-cut.

And to " measure " is to place a shorter measuring length s successively ( q times ) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q * s, q being the quotient, or remainder r is the " modulus ", the integer-fractional part left over after the division.

This division has been used here, although it is not clear whether the approach will be upheld by future research as some of the clades are weakly supported.

Since the evolution of the Aramaic alphabet out of the Phoenician one was a gradual process, the division of the world's alphabets into those derived from the Phoenician one directly and those derived from Phoenician via Aramaic is somewhat artificial.

By preserving the value, only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.

Eli Whitney is sometimes credited with developing the armory system of manufacturing in 1801, using the ideas of division of labor, engineering tolerance, and interchangeable parts to create assemblies from parts in a repeatable manner.

However, the team could not sustain its status and is now back in the second division.