Natural differences and integers

Differences between natural and whole:

The set of integers it is not a set well-ordered (since the subset of negative integers has no minimum)

Any subset not empty of \mathbb{Z} lower bound has minimum (and multiplied by (-1), if bound above, has maximum)

To multiply integers we have to distinguish between positive and negative, applying the so called rule of the signs:

\begin{cases} \text{positive }\cdot\text{positive = positive} \\ \text{positive }\cdot\text{negative = negative} \\ \text{negative }\cdot\text{positive = negative} \\ \text{negative }\cdot\text{negative = positive} \end{cases}

The order of \mathbb{Z} is a total order, but it should be noted that any negative number is less than any positive. In addition, these properties must be taken into account for operations:

\tiny\begin{cases} a \leq b \Rightarrow a + c \leq b + c \\ a \leq b, c \geq 0 \Rightarrow a \cdot c \leq b \cdot c \\ a \leq b, c < 0 \Rightarrow a \cdot c \geq b \cdot c & \text{if multiplied by a negative number} \end{cases}

We have at our disposal the absolute value function (or module):

\|a\|=\begin{cases} a & \text{if } a \geq 0 \\ (-a) & \text{if } a < 0 \end{cases}

From the absolute value function and thanks to its geometric implications, we also have at our disposal the Triangular inequality: