# 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)β $\frac{a}{b}$β of $a$ and $b$ may be defined as such a number (or element of the field) $x$ thatβ $b\beta \x8b\x85x=a$.β Thus,

$$b\beta \x8b\x85\frac{a}{b}=a,$$ |

which is the βfundamental property of quotientβ.

The quotient of the numbers $a$ and $b$ ($\beta \x890$) is a uniquely determined number, since if one had

$$\frac{a}{b}=x\beta \x89y=\frac{a}{b},$$ |

then we could write

$$b\beta \x81\u2019(x-y)=b\beta \x81\u2019x-b\beta \x81\u2019y=a-a=0$$ |

from which the supposition $b\beta \x890$ would imply $x-y=0$, i.e. $x=y$.

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

$$\frac{a}{b}={b}^{-1}\beta \x8b\x85a$$ |

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

$$b\beta \x81\u2019({b}^{-1}\beta \x81\u2019a)=(b\beta \x81\u2019{b}^{-1})\beta \x81\u2019a=1\beta \x81\u2019a=a.$$ |

- β’
For positive numbers the quotient may be obtained by performing the division algorithm with $a$ and $b$.β Ifβ $a>b>0$,β then $\frac{a}{b}$ indicates how many times $b$ fits in $a$.

- β’
The quotient of $a$ and $b$ does not change if both numbers (elements) are multiplied (or divided, which action is called reduction) by any β$k\beta \x890$:

$$\frac{k\beta \x81\u2019a}{k\beta \x81\u2019b}={(k\beta \x81\u2019b)}^{-1}\beta \x81\u2019(k\beta \x81\u2019a)={b}^{-1}\beta \x81\u2019{k}^{-1}\beta \x81\u2019k\beta \x81\u2019a={b}^{-1}\beta \x81\u2019a=\frac{a}{b}$$ So we have the method for getting the quotient of complex numbers,

$$\frac{a}{b}=\frac{\stackrel{{\rm B}\u2015}{b}\beta \x81\u2019a}{\stackrel{{\rm B}\u2015}{b}\beta \x81\u2019b},$$ where $\stackrel{{\rm B}\u2015}{b}$ is the complex conjugate of $b$, and the quotient of square root polynomials, e.g.

$$\frac{1}{5+2\beta \x81\u2019\sqrt{2}}=\frac{5-2\beta \x81\u2019\sqrt{2}}{(5-2\beta \x81\u2019\sqrt{2})\beta \x81\u2019(5+2\beta \x81\u2019\sqrt{2})}=\frac{5-2\beta \x81\u2019\sqrt{2}}{25-8}=\frac{5-2\beta \x81\u2019\sqrt{2}}{17};$$ in the first case one aspires after a real and in the second case after a rational denominator.

- β’
The division is neither associative nor commutative, but it is right distributive over addition:

$$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$$

See also the proportion equation.

## Comments

## The entry "division" became invisible.

The reason was a dollar-sign error =o)

Corrected!

BTW, the PM search engine has long been out of order. Itβs quite difficult to find entries on a wanted subject.

## Carmichael numbers and Devaraj numbers

Carmichael numbers constitute a sub-set of Devaraj numbers. To understand more about them refer sequences A 104016, A 104017 and A 162290. Some interesting facts pertaining to them will follow.