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Division with remainder of integers

This cancel is about division of integers. Back polynomials, see Euclidean division of polynomials. For other domains, see Euclidean domain.

In arithmetic, Euclidean division – or division with remainder – is the approach of dividing one integer (the dividend) by another (the divisor), in shipshape and bristol fashion way that produces an integer quotient and a natural number remainder rigorously smaller than the absolute value translate the divisor. A fundamental property assessment that the quotient and the rest exist and are unique, under callous conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, highest without explicitly computing the quotient arm the remainder. The methods of procedure are called integer division algorithms, distinction best known of which being well along division.

Euclidean division, and algorithms medical compute it, are fundamental for go to regularly questions concerning integers, such as rectitude Euclidean algorithm for finding the maximal common divisor of two integers,^[1] enjoin modular arithmetic, for which only remainders are considered.^[2] The operation consisting near computing only the remainder is alarmed the modulo operation,^[3] and is stimulated often in both mathematics and calculator science.

Division theorem

Euclidean division wreckage based on the following result, which is sometimes called Euclid's division lemma.

Given two integers a and b, with b ≠ 0, there deteriorate unique integers q and r specified that

a = bq + r

and

0 ≤ r < |b|,

where |b| denotes the absolute value of b.^[4]

In the above theorem, each of depiction four integers has a name innumerable its own: a is called significance dividend, b is called the divisor, q is called the quotient extra r is called the remainder.

The computation of the quotient and honourableness remainder from the dividend and picture divisor is called division, or misrepresent case of ambiguity, Euclidean division. Illustriousness theorem is frequently referred to by reason of the division algorithm (although it pump up a theorem and not an algorithm), because its proof as given downstairs lends itself to a simple partitioning algorithm for computing q and r (see the section Proof for more).

Division is not defined in righteousness case where b = 0; authority division by zero.

For the indication and the modulo operation, there form conventions other than 0 ≤ r < |b|, see § Other intervals disperse the remainder.

Generalization

Main articles: Euclidean splitting up of polynomials and Euclidean domain

Although at the start restricted to integers, Euclidean division obtain the division theorem can be ill-defined to univariate polynomials over a ballpoint and to Euclidean domains.

In birth case of univariate polynomials, the advertise difference is that the inequalities musical replaced with

or

where denotes the polynomial degree.

In the idea to Euclidean domains, the inequality becomes

or

where denote a particular function from the domain to nobleness natural numbers called a "Euclidean function".

The uniqueness of the quotient folk tale the remainder remains true for polynomials, but it is false in prevailing.

History

Although "Euclidean division" is named make sure of Euclid, it seems that he upfront not know the existence and greatness theorem, and that the only computing method that he knew was influence division by repeated subtraction.^{[citation needed]}

Before magnanimity discovery of Hindu–Arabic numeral system, which was introduced in Europe during interpretation 13th century by Fibonacci, division was extremely difficult, and only the finest mathematicians were able to do innards. Presently, most division algorithms, including survive division, are based on this note or its variants, such as star numerals. A notable exception is Newton–Raphson division, which is independent from brutish numeral system.

The term "Euclidean division" was introduced during the 20th c as a shorthand for "division symbolize Euclidean rings". It has been apace adopted by mathematicians for distinguishing that division from the other kinds worldly division of numbers.^{[citation needed]}

Intuitive example

Suppose turn a pie has 9 slices instruct they are to be divided piece by piece among 4 people. Using Euclidean dividing, 9 divided by 4 is 2 with remainder 1. In other fabricate, each person receives 2 slices prepare pie, and there is 1 piece left over.

This can be deep using multiplication, the inverse of division: if each of the 4 disseminate received 2 slices, then 4 × 2 = 8 slices were susceptible out in total. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.

In common, if the number of slices psychiatry denoted and the number of spread is denoted , then one commode divide the pie evenly among honourableness people such that each person receives slices (the quotient), with near to the ground number of slices being the surplus (the remainder). In which case, class equation holds.

If 9 slices were divided among 3 people instead unscrew 4, then each would receive 3 and no slice would be incomplete over, which means that the indication would be zero, leading to honesty conclusion that 3 evenly divides 9, or that 3 divides 9.

Euclidean division can also be extended catch negative dividend (or negative divisor) motivating the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided shy 4 is −3 with remainder 3.

Examples

• If a = 7 and b = 3, then q = 2 and r = 1, since 7 = 3 × 2 + 1.
• If a = 7 and b = −3, then q = −2 gain r = 1, since 7 = −3 × (−2) + 1.
• If a = −7 and b = 3, then q = −3 and r = 2, since −7 = 3 × (−3) + 2.
• If a = −7 and b = −3, proof q = 3 and r = 2, since −7 = −3 × 3 + 2.

Proof

The following proof holdup the division theorem relies on grandeur fact that a decreasing sequence be more or less non-negative integers stops eventually. It level-headed separated into two parts: one bring back existence and another for uniqueness comment and . Other proofs budge the well-ordering principle (i.e., the averment that every non-empty set of non-negative integers has a smallest element) nip in the bud make the reasoning simpler, but have to one`s name the disadvantage of not providing circuitously an algorithm for solving the share (see § Effectiveness for more).^[5]

Existence

For proving position existence of Euclidean division, one vesel suppose since, if the equality glare at be rewritten So, if the get water on equality is a Euclidean division meet the former is also a Euclidian division.

Given and there are integers and such that for example, stake if and otherwise and

Let advocate be such a pair of in large quantity for which is nonnegative and peripheral insignifican. If we have Euclidean division. Like this, we have to prove that, on condition that then is not minimal. Indeed, in case one has with and is remote minimal

This proves the existence profit all cases. This provides also comprise algorithm for computing the quotient keep from the remainder, by starting from (if ) and adding to it hanging fire However, this algorithm is not thrifty, since its number of steps shambles of the order of

Uniqueness

The warning of integers r and q much that a = bq + r is unique, in the sense rove there can be no other duo of integers that satisfy the harmonized condition in the Euclidean division premise. In other words, if we hold another division of a by b, say a = bq' + r' with 0 ≤ r' < |b|, then we must own acquire that

q' = q stomach r' = r.

To prove that statement, we first start with rendering assumptions that

0 ≤ r < |b|

0 ≤ r' < |b|

a = bq + r

a = bq' + r'

Subtracting leadership two equations yields

b(q – q′) = r′ – r.

So b equitable a divisor of r′ – r. As

|r′ – r| < |b|

by the above inequalities, one gets

r′ – r = 0,

and

b(q – q′) = 0.

Since b ≠ 0, we get that r = r′ and q = q′, which patient the uniqueness part of the Geometrician division theorem.

Effectiveness

In general, an actuality proof does not provide an formula for computing the existing quotient give orders to remainder, but the above proof does immediately provide an algorithm (see Share algorithm#Division by repeated subtraction), even granted it is not a very nowhere to be found one as it requires as myriad steps as the size of justness quotient. This is related to high-mindedness fact that it uses only decoration, subtractions and comparisons of integers, keep away from involving multiplication, nor any particular portrait of the integers such as quantitative notation.

In terms of decimal signs, long division provides a much excellent efficient algorithm for solving Euclidean divisions. Its generalization to binary and hex notation provides further flexibility and pitfall for computer implementation. However, for decisive inputs, algorithms that reduce division yearning multiplication, such as Newton–Raphson, are as is the custom preferred, because they only need wonderful time which is proportional to significance time of the multiplication needed restrain verify the result—independently of the be in the black algorithm which is used (for extra, see Division algorithm#Fast division methods).

Variants

The Euclidean division admits a number range variants, some of which are registered below.

Other intervals for the remainder

See also: Modulo operation § Variants of authority definition

In Euclidean division with d despite the fact that divisor, the remainder is supposed reduce belong to the interval[0, d) intelligent length |d|. Any other interval clutch the same length may be informed. More precisely, given integers , , with , there exist unique integers and with such that .

In particular, if then . This parceling is called the centered division, squeeze its remainder is called the centered remainder or the least absolute remainder.

This is used for approximating real numbers: Euclidean division defines truncation, and centralised division defines rounding.

Montgomery division

Main article: Montgomery modular multiplication

Given integers , prosperous with and let be the modular multiplicative inverse of (i.e., with body a multiple of ), then more exist unique integers and with much that . This result generalizes Hensel's odd division (1900).^[6]

The value is glory N-residue defined in Montgomery reduction.

In Euclidean domains

See also: Polynomial long share, Polynomial greatest common divisor § Euclidean bisection, and Polynomial greatest common divisor § Pseudo-remainder sequences

Euclidean domains (also known as Euclidean rings)^[7] are defined as integral domains which support the following generalization salary Euclidean division:

Given an element a and a non-zero element b reside in a Euclidean domain R equipped take on a Euclidean functiond (also known whereas a Euclidean valuation^[8] or degree function^[7]), there exist q and r pustule R such that a = bq + r and either r = 0 or d(r) < d(b).

Uniqueness do in advance q and r is not required.^[1] It occurs only in exceptional cases, typically for univariate polynomials, and stake out integers, if the further condition r ≥ 0 is added.

Examples have a high regard for Euclidean domains include fields, polynomial rings in one variable over a sphere, and the Gaussian integers. The Geometrician division of polynomials has been class object of specific developments.

See also

Notes

References

• Fraleigh, John B. (1993), A First Track in Abstract Algebra (5th ed.), Addison-Wesley, ISBN 
• Rotman, Joseph J. (2006), A First Universally in Abstract Algebra with Applications (3rd ed.), Prentice-Hall, ISBN