In order for the operation of subtraction to be always possible, we have the extend the natural numbers by zero ( the “neutral” element of addition) and the negative numbers to the set $\mathbf{Z}$ of whole numbers

The introduction of the zero as a number was historically not a trivial step. Zero is connected with the notion of NOTHING, and for the pre-socratic natural philosophers it was a fundamental question, whether NOTHING ( the emptiness, something that is not ) can exist or not.

Parmenides of Elea (around 600 b.C.) taught, that NOTHING cannot exist, but that every space has to filled by something, which leads to the paradoxical logical consequence, that movement is impossible and everything is unchangeable. The atomicists Leukipp (5th century b.C.) and Democrit (460 – 371 b.C.) taught however, that the world consists mostly of nothing (today we would call it vacuum), in which objects that consist of material atoms can move.

In the 3rd century b.C. the zero was imported from the east in connection with the campaigns of Alexander the great as a sign indicating the position in the decimal number system (as today in 10,100). The meaning of a whole number zero only obtained in the 17th century.

The set of whole numbers contains the natural numbers as sub set

Whole numbers: ......,-3, -2, -1, 0, 1, 2, 3.........

Multiplication is admissible without exception if one defines: $\left(-1\right)\cdot 1=-1;\left(-1\right)\cdot \left(-1\right)=1$ and $\left(0\right)\cdot \left(1\right)=0$

In the domain of whole numbers the symbol for positive infinity must be necessarily supplemented by negative infinity $-\infty ;\infty $; both numbers are no numbers in the usual sense.

Division can be applied for whole numbers as for the natural numbers, if the divisor is contain as a factor in the dividend, i.e. if the division works out, as for $-30:5=-6$.

Division by zero is not a well defined inversion of multiplication:

$$\text{forinteger}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}a,b,c$$

$$\frac{b}{a}=c\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{uniquelyleadsto}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}a=\frac{b}{c}$$

$$\text{for}\phantom{\rule{0em}{0ex}}\frac{0}{a}=0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}a\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\text{canbeanynumber}\phantom{\rule{0em}{0ex}},$$

and is therefore excluded. The expressions $0\cdot \infty ,\frac{\infty}{\infty}$ and $\frac{0}{0}$ are not defined.

Whole numbers are visualized as a discrete ladder on the number line (see Figure 3.2). Arithmetic operations amount to jumping back and forth on this ladder – such as toddlers indeed make calculations with natural numbers by counting.

$$-5\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}-4\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}-3\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}-2\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}-1\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}0\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}1\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}2\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}3\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}4\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}5$$

$$\left|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\right|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\right|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\right|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}$$