Difference between revisions of "Remainder"

m
 
Line 1: Line 1:
 +
{{stub}}
 +
 
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.
  
Line 6: Line 8:
  
 
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>).
 
{{stub}}
 

Latest revision as of 17:01, 7 November 2008

This article is a stub. Help us out by expanding it.

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, $q = 1.\overline{3}$).

Invalid username
Login to AoPS