Difference between revisions of "Remainder"

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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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Revision as of 18:43, 17 June 2008

The remainder of a division of two integers $\frac {a}{b},\ b \neq 0$ is the integer $r < b$ such that $a = qb + r$, where $q$ is the quotient; in other words, $r$ is the part of $a$ that is not divisible by $b$. If $a = 4$, and $b = 3$, for example, the division $\frac {4}{3}$ would have remainder $1$, since $4 = (1)3 + 1$ (notice that the quotient, in this case, is one). If $b$ is a divisor of $a$, the remainder is said to be zero.


The concept of a remainder is related to modular arithmetic: $r$ is said to be the residue class of $a$ in modulo $b$ iff $a = qb + r$ (an equivalent statement would be $a \equiv r \mod b$).


It is important to notice that the remainder is most useful when an integer quotient is desired, as we can always say that $a = qb$ for any real number $q$ (in the example provided earlier, $q = 1.\overline{3}$).

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