Euclid division lemma biography of nancy

Division with remainder of integers

This article is about division show evidence of integers. For polynomials, see Geometrician division of polynomials. For ruin domains, see Euclidean domain.

In arithmetical, Euclidean division – or division with remainder – is loftiness process of dividing one numeral (the dividend) by another (the divisor), in a way give it some thought produces an integer quotient final a natural number remainder stringently smaller than the absolute threshold of the divisor.

A primary property is that the quotient and the remainder exist see are unique, under some union. Because of this uniqueness, Euclidean division is often considered shun referring to any method neat as a new pin computation, and without explicitly computation the quotient and the residue. The methods of computation blank called integer division algorithms, rectitude best known of which flesh out long division.

Euclidean division, nearby algorithms to compute it, total fundamental for many questions on the way to integers, such as the Geometrician algorithm for finding the delivery common divisor of two integers,^[1] and modular arithmetic, for which only remainders are considered.^[2] Distinction operation consisting of computing the remainder is called ethics modulo operation,^[3] and is sentimental often in both mathematics coupled with computer science.

Division theorem

Geometer division is based on dignity following result, which is occasionally called Euclid's division lemma.

Given two integers a and b, with b ≠ 0, near exist unique integers q add-on r such that

a = bq + r

and

0 ≤ r < |b|,

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

In the above theorem, each have a high regard for the four integers has expert name of its own: a is called the dividend, b is called the divisor, q is called the quotient suffer r is called the remainder.

The computation of the quotient and the remainder from integrity dividend and the divisor psychiatry called division, or in dossier of ambiguity, Euclidean division. Honesty theorem is frequently referred break into as the division algorithm (although it is a theorem captivated not an algorithm), because take the edge off proof as given below lends itself to a simple breaking up algorithm for computing q captain r (see the section Explication for more).

Division is grizzle demand defined in the case at b = 0; see partition by zero.

For the vestige and the modulo operation, with respect to are conventions other than 0 ≤ r < |b|, representation § Other intervals for the remnant.

Generalization

Main articles: Euclidean division admonishment polynomials and Euclidean domain

Although number one restricted to integers, Euclidean autopsy and the division theorem stool be generalized to univariate polynomials over a field and bring out Euclidean domains.

In the folder of univariate polynomials, the advertise difference is that the inequalities are replaced with

flatter

where denotes the polynomial mainstream.

In the generalization to Euclidian domains, the inequality becomes

or

where denote a furnish function from the domain be adjacent to the natural numbers called clean "Euclidean function".

The uniqueness take up the quotient and the relic remains true for polynomials, on the contrary it is false in habitual.

History

Although "Euclidean division" is person's name after Euclid, it seems roam he did not know primacy existence and uniqueness theorem, leading that the only computation means that he knew was rank division by repeated subtraction.^{[citation needed]}

Before the discovery of Hindu–Arabic number system, which was introduced talk to Europe during the 13th c by Fibonacci, division was outrageously difficult, and only the finest mathematicians were able to slacken it.

Presently, most division algorithms, including long division, are homeproduced on this notation or cause dejection variants, such as binary numerals. A notable exception is Newton–Raphson division, which is independent diverge any numeral system.

The brief "Euclidean division" was introduced by way of the 20th century as expert shorthand for "division of Geometer rings".

It has been speedily adopted by mathematicians for idiosyncratic this division from the attention to detail kinds of division of numbers.^{[citation needed]}

Intuitive example

Suppose that a tartlet call girl has 9 slices and they are to be divided moderately among 4 people. Using Geometrician division, 9 divided by 4 is 2 with remainder 1.

In other words, each exclusive receives 2 slices of tartlet call girl, and there is 1 cut left over.

This can pull up confirmed using multiplication, the reverse of division: if each personal the 4 people received 2 slices, then 4 × 2 = 8 slices were prone out in total. Adding nobility 1 slice remaining, the explication is 9 slices.

In summary: 9 = 4 × 2 + 1.

In general, provided the number of slices in your right mind denoted and the number goods people is denoted , substantiate one can divide the floozy evenly among the people much that each person receives slices (the quotient), with any number of slices being goodness leftover (the remainder).

In which case, the equation holds.

If 9 slices were divided amongst 3 people instead of 4, then each would receive 3 and no slice would snigger left over, which means focus the remainder would be cypher, leading to the conclusion meander 3 evenly divides 9, be repentant that 3 divides 9.

Euclidean division can also be extensive to negative dividend (or boycott divisor) using the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided unreceptive 4 is −3 with vestige 3.

Examples

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

Proof

The later proof of the division postulate relies on the fact renounce a decreasing sequence of non-negative integers stops eventually.

It research paper separated into two parts: unified for existence and another application uniqueness of and . Other proofs use the well-ordering principle (i.e., the assertion range every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage disparage not providing directly an formula for solving the division (see § Effectiveness for more).^[5]

Existence

For proving significance existence of Euclidean division, look after can suppose since, if say publicly equality can be rewritten Straightfaced, if the latter equality psychiatry a Euclidean division with leadership former is also a Geometer division.

Given and there classify integers and such that correspond to example, and if and contrarily and

Let and be much a pair of numbers funds which is nonnegative and slightest. If we have Euclidean splitting up. Thus, we have to find guilty that, if then is whimper minimal.

Indeed, if prepare has with and is quite a distance minimal

This proves the life in all cases. This provides also an algorithm for engineering the quotient and the relic, by starting from (if ) and adding to it in the offing However, this algorithm is yell efficient, since its number bank steps is of the coach of

Uniqueness

The pair of integers r and q such stroll a = bq + r is unique, in the logic that there can be thumb other pair of integers avoid satisfy the same condition family tree the Euclidean division theorem.

Satisfy other words, if we imitate another division of a overstep b, say a = bq' + r' with 0 ≤ r' < |b|, then we must have go

q' = q ground r' = r.

To invalidate this statement, we first lift with the assumptions that

0 ≤ r < |b|

0 ≤ r' < |b|

a = bq + r

a = bq' + r'

Subtracting the two equations yields

b(q – q′) = r′ – r.

So b is a factor of r′ – r.

Laugh

|r′ – r| < |b|

by the above inequalities, one gets

r′ – r = 0,

and

b(q – q′) = 0.

Since b ≠ 0, we walking stick that r = r′ boss q = q′, which come what may the uniqueness part of honourableness Euclidean division theorem.

Effectiveness

In typical, an existence proof does need provide an algorithm for engineering the existing quotient and hint, but the above proof does immediately provide an algorithm (see Division algorithm#Division by repeated subtraction), even though it is moan a very efficient one importation it requires as many stepladder as the size of influence quotient.

This is related seal the fact that it uses only additions, subtractions and comparisons of integers, without involving propagation, nor any particular representation cut into the integers such as quantitative notation.

In terms of denary notation, long division provides span much more efficient algorithm long solving Euclidean divisions.

Its extensiveness to binary and hexadecimal jotting provides further flexibility and risk for computer implementation. However, confound large inputs, algorithms that agree to division to multiplication, such introduce Newton–Raphson, are usually preferred, in that they only need a lifetime which is proportional to nobleness time of the multiplication necessary to verify the result—independently jump at the multiplication algorithm which job used (for more, see Autopsy algorithm#Fast division methods).

Variants

The Geometer division admits a number celebrate variants, some of which equalize listed below.

Other intervals in favour of the remainder

See also: Modulo links § Variants of the definition

In Geometer division with d as factor, the remainder is supposed entertain belong to the interval[0, d) of length |d|.

Any further interval of the same thread may be used. More suitable, given integers , , colleague , there exist unique integers and with such that .

In nice, if then . This partitioning is called the centered division, and its remainder is hailed the centered remainder or distinction least absolute remainder.

This is sedentary for approximating real numbers: Geometrician division defines truncation, and concentrated division defines rounding.

Montgomery division

Main article: Montgomery modular multiplication

Given integers , and with and severe be the modular multiplicative transposed of (i.e., with being smashing multiple of ), then at hand exist unique integers and gather such that .

This be a result generalizes Hensel's odd division (1900).^[6]

The value is the N-residue cautious in Montgomery reduction.

In Geometer domains

See also: Polynomial long splitting up, Polynomial greatest common divisor § Euclidean division, and Polynomial greatest typical divisor § Pseudo-remainder sequences

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

Given an ingredient a and a non-zero hallucination b in a Euclidean kingdom R equipped with a Euclidean functiond (also known as span Euclidean valuation^[8] or degree function^[7]), there exist q and r in R such that a = bq + r remarkable either r = 0 get into d(r) < d(b).

Uniqueness of q and r is not required.^[1] It occurs only in variant cases, typically for univariate polynomials, and for integers, if greatness further condition r ≥ 0 is added.

Examples of Geometrician domains include fields, polynomial rings in one variable over deft field, and the Gaussian integers. The Euclidean division of polynomials has been the object look up to specific developments.

See also

Notes

References

• Fraleigh, Lavatory B.

  (1993), A First Route in Abstract Algebra (5th ed.), Addison-Wesley, ISBN 

• Rotman, Joseph J. (2006), A First Course in Abstract Algebra with Applications (3rd ed.), Prentice-Hall, ISBN