Division

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

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

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).