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The operation inverse to [[Multiplication|multiplication]]: To find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336701.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336702.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336703.png" /> for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336705.png" />. The result <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336706.png" /> of the division is known as the quotient or the ratio between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336708.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d0336709.png" /> is the divided, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367010.png" /> is the divisor. The operation of division is denoted by a colon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367011.png" />, a horizontal stroke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367012.png" /> or an oblique stroke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367013.png" />.
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The operation inverse to [[Multiplication|multiplication]]: To find an $x$ such that $bx=a$ or $xb=a$ for given $a$ and $b$. The result $x$ of the division is known as the quotient or the ratio between $a$ and $b$; $a$ is the dividend, while $b$ is the divisor. The operation of division is denoted by a colon $(a:b)$, a horizontal stroke $\frac ab$ or an oblique stroke $(a/b)$.
  
In the field of rational numbers, division (except for division by zero) is always possible, and the result of a division is unique. In the ring of integers division is not always possible. Thus, 10 is divisible by 5, but is not divisible by 3. If the division of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367014.png" /> by an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367015.png" /> in the field of rational numbers yields a quotient which is also an integer, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367016.png" /> is totally divisible (divisible without remainder) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367017.png" />; this is noted as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367018.png" />. Division of complex numbers is defined by the formula
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In the field of rational numbers, division (except for division by zero) is always possible, and the result of a division is unique. In the ring of integers division is not always possible. Thus, 10 is divisible by 5, but is not divisible by 3. If the division of an integer $a$ by an integer $b$ in the field of rational numbers yields a quotient which is also an integer, one says that $a$ is totally divisible (divisible without remainder) by $b$; this is noted as $b\mid a$. Division of complex numbers is defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367019.png" /></td> </tr></table>
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\[\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2},\]
  
 
while division of the complex numbers in their trigonometric form is given by the formula
 
while division of the complex numbers in their trigonometric form is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367020.png" /></td> </tr></table>
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\[\frac{r(\cos\alpha+i\sin\alpha)}{\rho(\cos\beta+i\sin\beta)}=\frac r\rho(\cos(\alpha-\beta)+i\sin(\alpha-\beta)).\]
  
Division with remainder is actually a separate operation, which is different from division as defined above. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367022.png" /> are integers, then division with remainder of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367023.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367024.png" /> consists of finding integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367026.png" /> such that
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Division with remainder is actually a separate operation, which is different from division as defined above. If $a$ and $b\neq0$ are integers, then division with remainder of $a$ by $b$ consists of finding integers $x$ and $y$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367027.png" /></td> </tr></table>
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\[a=bx+y,\quad\text{where }0\leq y\leq|b|.\]
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367028.png" /> is the divided, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367029.png" /> is the divisor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367030.png" /> is the quotient, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367031.png" /> is the remainder. This operation is always possible and is unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367032.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367033.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367034.png" /> without remainder. The quotient will then be the same as in ordinary division.
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Here $a$ is the dividend, $b$ is the divisor, $x$ is the quotient, and $y$ is the remainder. This operation is always possible and is unique. If $y=0$, one says that $b$ divides $a$ without remainder. The quotient will then be the same as in ordinary division.
  
Division with remainder of polynomials with coefficients in a given field is defined in a similar manner. It consists in finding, for two given polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367036.png" />, polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367038.png" /> satisfying the conditions
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Division with remainder of polynomials with coefficients in a given field is defined in a similar manner. It consists in finding, for two given polynomials $A(x)$ and $B(x)$, polynomials $Q(x)$ and $R(x)$ satisfying the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367039.png" /></td> </tr></table>
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\[A(x)=B(x)Q(x)+R(x),\]
  
where the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367040.png" /> is less than that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367041.png" />. This operation is also always possible and is unique. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367042.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367043.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367044.png" /> without remainder.
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where the degree of $R(x)$ is less than that of $Q(x)$. This operation is also always possible and is unique. If $R(x)\equiv0$, one says that $A(x)$ is divisible by $B(x)$ without remainder.
  
  
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Division (with remainder) is related to the [[Euclidean algorithm|Euclidean algorithm]].
 
Division (with remainder) is related to the [[Euclidean algorithm|Euclidean algorithm]].
  
Division of a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367045.png" /> by a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367046.png" /> amounts to multiplying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367047.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367048.png" /> and dividing by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367049.png" />, i.e.
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Division of a complex number $z$ by a complex number $w\neq0$ amounts to multiplying $z$ by $\bar w$ and dividing by $|w|^2$, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367050.png" /></td> </tr></table>
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\[\frac zw=\frac{z\bar w}{|w|^2}.\]
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367051.png" /> is the complex conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367053.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033670/d03367054.png" /> (cf. [[Complex number|Complex number]]).
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Here, $\bar w$ is the complex conjugate of $w$ and $|w|$ is the norm of $w$ (cf. [[Complex number|Complex number]]).

Revision as of 18:39, 30 December 2018

The operation inverse to multiplication: To find an $x$ such that $bx=a$ or $xb=a$ for given $a$ and $b$. The result $x$ of the division is known as the quotient or the ratio between $a$ and $b$; $a$ is the dividend, while $b$ is the divisor. The operation of division is denoted by a colon $(a:b)$, a horizontal stroke $\frac ab$ or an oblique stroke $(a/b)$.

In the field of rational numbers, division (except for division by zero) is always possible, and the result of a division is unique. In the ring of integers division is not always possible. Thus, 10 is divisible by 5, but is not divisible by 3. If the division of an integer $a$ by an integer $b$ in the field of rational numbers yields a quotient which is also an integer, one says that $a$ is totally divisible (divisible without remainder) by $b$; this is noted as $b\mid a$. Division of complex numbers is defined by the formula

\[\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2},\]

while division of the complex numbers in their trigonometric form is given by the formula

\[\frac{r(\cos\alpha+i\sin\alpha)}{\rho(\cos\beta+i\sin\beta)}=\frac r\rho(\cos(\alpha-\beta)+i\sin(\alpha-\beta)).\]

Division with remainder is actually a separate operation, which is different from division as defined above. If $a$ and $b\neq0$ are integers, then division with remainder of $a$ by $b$ consists of finding integers $x$ and $y$ such that

\[a=bx+y,\quad\text{where }0\leq y\leq|b|.\]

Here $a$ is the dividend, $b$ is the divisor, $x$ is the quotient, and $y$ is the remainder. This operation is always possible and is unique. If $y=0$, one says that $b$ divides $a$ without remainder. The quotient will then be the same as in ordinary division.

Division with remainder of polynomials with coefficients in a given field is defined in a similar manner. It consists in finding, for two given polynomials $A(x)$ and $B(x)$, polynomials $Q(x)$ and $R(x)$ satisfying the conditions

\[A(x)=B(x)Q(x)+R(x),\]

where the degree of $R(x)$ is less than that of $Q(x)$. This operation is also always possible and is unique. If $R(x)\equiv0$, one says that $A(x)$ is divisible by $B(x)$ without remainder.


Comments

Division (with remainder) is related to the Euclidean algorithm.

Division of a complex number $z$ by a complex number $w\neq0$ amounts to multiplying $z$ by $\bar w$ and dividing by $|w|^2$, i.e.

\[\frac zw=\frac{z\bar w}{|w|^2}.\]

Here, $\bar w$ is the complex conjugate of $w$ and $|w|$ is the norm of $w$ (cf. Complex number).

How to Cite This Entry:
Division. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division&oldid=15710
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article