Difference between revisions of "Remainder"
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The '''remainder''' of a division of two integers <math>\frac {a}{b},\ b \neq 0</math> is the integer <math>r < b</math> such that <math>a = qb + r</math>, where <math>q</math> is the [[Division|quotient]]; in other words, <math>r</math> is the part of <math>a</math> that is not [[Divisibility|divisible]] by <math>b</math>. If <math>a = 4</math>, and <math>b = 3</math>, for example, the division <math>\frac {4}{3}</math> would have remainder <math>1</math>, since <math>4 = (1)3 + 1</math> (notice that the quotient, in this case, is one). If <math>b</math> is a [[divisor]] of <math>a</math>, the remainder is said to be zero. | The '''remainder''' of a division of two integers <math>\frac {a}{b},\ b \neq 0</math> is the integer <math>r < b</math> such that <math>a = qb + r</math>, where <math>q</math> is the [[Division|quotient]]; in other words, <math>r</math> is the part of <math>a</math> that is not [[Divisibility|divisible]] by <math>b</math>. If <math>a = 4</math>, and <math>b = 3</math>, for example, the division <math>\frac {4}{3}</math> would have remainder <math>1</math>, since <math>4 = (1)3 + 1</math> (notice that the quotient, in this case, is one). If <math>b</math> is a [[divisor]] of <math>a</math>, the remainder is said to be zero. | ||
The concept of a remainder is related to [[modular arithmetic]]: <math>r</math> is said to be the [[residue class]] of <math>a</math> in modulo <math>b</math> [[iff]] <math>a = qb + r</math> (an equivalent statement would be <math>a \equiv r \mod b</math>). | The concept of a remainder is related to [[modular arithmetic]]: <math>r</math> is said to be the [[residue class]] of <math>a</math> in modulo <math>b</math> [[iff]] <math>a = qb + r</math> (an equivalent statement would be <math>a \equiv r \mod b</math>). | ||
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It is important to notice that the remainder is most useful when an integer quotient is desired, as we can always say that <math>a = qb</math> for any [[real number]] <math>q</math> (in the example provided earlier, <math>q = 1.\overline{3}</math>). | It is important to notice that the remainder is most useful when an integer quotient is desired, as we can always say that <math>a = qb</math> for any [[real number]] <math>q</math> (in the example provided earlier, <math>q = 1.\overline{3}</math>). | ||
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Latest revision as of 16:01, 7 November 2008
This article is a stub. Help us out by expanding it.
The remainder of a division of two integers 
is the integer 
such that 
, where 
is the quotient; in other words, 
is the part of 
that is not divisible by 
. If 
, and 
, for example, the division 
would have remainder 
, since 
(notice that the quotient, in this case, is one). If is a divisor of , the remainder is said to be zero.
The concept of a remainder is related to modular arithmetic: is said to be the residue class of in modulo iff (an equivalent statement would be 
).
It is important to notice that the remainder is most useful when an integer quotient is desired, as we can always say that 
for any real number (in the example provided earlier, 
).
