### 3.3 Rational Numbers

In order for division to admissible in all cases, except division by
$z$, the
whole numbers have to supplemented by the “broken numbers” to the set of rational
numbers $\mathbf{Q}$
Rational numbers contain the whole numbers as sub set

$$\text{rationalnumber}=\text{wholenumber}:\text{wholenumber}=\frac{\text{wholenumber}}{\text{wholenumber}}$$

$$\text{Examples:}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}-5;-\frac{3}{2};1175\u22151176;3;1.1357;5.28666666\cdots ;\cdots $$
When written as a decimal number rational numbers are decimal numbers
with a remainder that has a finite length or with periodically repeating
digits.

There is no largest rational number.

It is obvious, that whole numbers are rare special cases of rational numbers.
Between two subsequent whole numbers there are infinitely many rational
numbers

Division by zero is still not a well defined inversion of multiplication and remains
formally excluded. If one starts from the concept of zero as the limit of a sequence of
nearly infinitely small positive or negative rational numbers, then division by zero would
be equivalent to the definition of nearly infinite positive or negative numbers. In this
symbolic sense division by zero can be associated with a sequence that has a limit of
$\pm \infty $.

Taking the power of a number is defined for rational numbers as repeated multiplication with
whole numbers $n$
as exponent by

$${A}^{n}=A\cdot A\cdot A\cdot A\cdot A\cdots n\text{times}$$

$${A}^{0}=1;{A}^{-N}=\frac{1}{{A}^{n}}$$
Taking the $n$-th
root is the logical inversion of taking the power. In the domain of rational numbers
root taking is possible

- if the exponent $n$
of the root is odd
- or when for even root exponents the original number (the radicand, the
number under the root sign ) is positive.
- and if in both cases the operation results in rational numbers which is only
the case for rare radicands, that can be reduced to fractions of powers, for
example $\sqrt{6.25}=\sqrt{\frac{625}{100}}=\frac{25}{10}=2.5\phantom{\rule{0em}{0ex}}.$