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