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Homedivision

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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

$b(b^{{-1}}a)=(bb^{{-1}})a=1a=a.$ |

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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 action is called reduction

^{}) 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 square 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:$\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$

## Mathematics Subject Classification

00A05*no label found*12E99

*no label found*

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## 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. It’s 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.