Difference between revisions of "Remainder"

Latest revision as of 16:01, 7 November 2008

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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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-A remainder is the amount left over from division. For example, if you do 7 divided by 4, you have a remainder of 3. If you have 7 pencils and 4 friends, and give each friend 1 pencil, you will still have 3 pencils left, which is not divisible by 4.
+{{stub}}
-Remainders cannot be greater than or equal to the divisor. If the remainder is equal to the divisor, the remainder is 0.
+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>).
+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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Difference between revisions of "Remainder"

Latest revision as of 16:01, 7 November 2008