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division and algorithm
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.
The division algorithm ( see Euclidean division ) is a theorem expressing the outcome of division in the natural numbers and more general rings.
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.

division and is
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.

division and mathematical
They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction, square root and cubic root.
Blind students also complete mathematical assignments using a braille-writer and Nemeth code ( a type of braille code for mathematics ) but large multiplication and long division problems can be long and difficult.
Frets are laid out to a mathematical ratio that results in equal tempered division of the octave.
The Handbook of Mathematical Logic makes a rough division of contemporary mathematical logic into four areas:
Unlike Structuralists, however, the Post-structuralists questioned the division between relation and component and, correspondingly, did not attempt to reduce the subjects of their study to an essential set of relations that could be portrayed with abstract, functional schemes or mathematical symbols ( as in Claude Lévi-Strauss's algebraic formulation of mythological transformation in " The Structural Study of Myth ").
The various invalid mathematical proofs ( e. g., that 1 = 2 ) are classic examples, generally relying on a hidden division by zero.
* Herbert Daniel Landahl, PhD, Professor Emeritus of Biophysics and Mathematical Biology-Basic research in mathematical biophysics of the central nervous system, cell division dynamics, population interactions, and control of insulin bioynthesis.
* Division algorithm, a method for computing the result of mathematical division
Meanwhile, division was represented with the classic mathematical division symbol ( the obelus, ÷), which was implemented by overstriking a minus sign and a colon ( on both EBCDIC and ASCII paper terminals ).
** division, the mathematical operation that is the inverse of multiplication
Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.
It is mainly used to represent the mathematical operation of division.
A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined.
Today, inexpensive calculators and computers have become the most common way to solve division problems, eliminating a traditional mathematical exercise, and decreasing the educational opportunity to show how to do so by paper and pencil techniques.
Theoretical computer science ( TCS ) is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing and includes the theory of computation.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry.
Advances in the numeral system and mathematical notation eventually led to the discovery of mathematical operations such as addition, subtraction, multiplication, division, squaring, square root, and so forth.
During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm ( for use in multiplication and division ) and the trigonometric functions.
In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis.
The last code is obtained by a mathematical method ( dividing by 97 the umber formed by the first 13 digits, take the remainder from this division, and then the " complement to 97 ", that is the difference between 97 and the left-over of the division ): this gives the control key code.

0.741 seconds.