# division

Division is the operation which assigns to every two numbers (or more generally, elements of a field) $a$ and $b$ their quotient or ratio, provided that the latter, $b$, is distinct from zero.

The quotient (or ratio)  $\frac{a}{b}$  of $a$ and $b$ may be defined as such a number (or element of the field) $x$ that  $b\cdot x=a$.  Thus,

 $b\cdot\frac{a}{b}=a,$

which is the “fundamental property of quotient”.

The quotient of the numbers $a$ and $b$ ($\neq 0$) is a uniquely determined number, since if one had

 $\frac{a}{b}=x\neq y=\frac{a}{b},$

then we could write

 $b(x-y)=bx-by=a-a=0$

from which the supposition $b\neq 0$ would imply $x-y=0$, i.e. $x=y$.

The explicit general expression for $\frac{a}{b}$ is

 $\frac{a}{b}=b^{-1}\cdot a$

where $b^{-1}$ is the inverse number (the multiplicative inverse) of $a$, because

 $b(b^{-1}a)=(bb^{-1})a=1a=a.$
• For positive numbers the quotient may be obtained by performing the division algorithm with $a$ and $b$.  If  $a>b>0$,  then $\frac{a}{b}$ indicates how many times $b$ fits in $a$.

• The quotient of $a$ and $b$ does not change if both numbers (elements) are multiplied (or divided, which is called ) by any  $k\neq 0$:

 $\frac{ka}{kb}=(kb)^{-1}(ka)=b^{-1}k^{-1}ka=b^{-1}a=\frac{a}{b}$

So we have the method for getting the quotient of complex numbers,

 $\frac{a}{b}=\frac{\bar{b}a}{\bar{b}b},$

where $\bar{b}$ is the complex conjugate of $b$, and the quotient of http://planetmath.org/SquareRootOfSquareRootBinomialsquare root polynomials, e.g.

 $\frac{1}{5+2\sqrt{2}}=\frac{5-2\sqrt{2}}{(5-2\sqrt{2})(5+2\sqrt{2})}=\frac{5-2% \sqrt{2}}{25-8}=\frac{5-2\sqrt{2}}{17};$

in the first case one aspires after a real and in the second case after a rational denominator.

• The division is neither associative nor commutative, but it is right distributive over addition:

 $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
 Title division Canonical name Division Date of creation 2014-08-08 17:51:29 Last modified on 2014-08-08 17:51:29 Owner pahio (2872) Last modified by pahio (2872) Numerical id 29 Author pahio (2872) Entry type Definition Classification msc 00A05 Classification msc 12E99 Related topic InverseFormingInProportionToGroupOperation Related topic DivisionInGroup Related topic ConjugationMnemonic Related topic Difference2 Related topic UniquenessOfDivisionAlgorithmInEuclideanDomain Defines quotient Defines ratio Defines fundamental property of quotient Defines reduction