\begin{defn}
The set of positive integers is usually denoted by $\mathbb{N}$, i.e.
$\mathbb{N}=\{1,2,3,4,....\}$ and is called the set of \textbf{natural numbers}. In some cases the number zero is included as a natural number, but here we will use the symbol $\mathbb{N}_0$ to denote the integers 0, 1, 2 and up.
\end{defn}

Within this set of numbers it is possible to add and multiply numbers together. Arithmetic operations are denoted by $+$ for addition and $\cdot$ (or $\times$) for multiplication. A natural number can also be raised to the power of a natural number, e.g. $3^5=3\cdot 3\cdot 3\cdot 3\cdot 3$
or in general $m^n=m\cdot m \cdot \ldots \cdot m$ ($n$ times).\\

When stating general properties of the natural numbers one needs to use symbols to indicate that the property holds for an arbitrary number. It is not enough to just write the property for a few numbers. For example, to declare that one can interchange numbers in a sum, it is not enough to say $4+3=3+4$ but one must explicitly state "the addition operator has the property that any two natural numbers, $n,\ m\in \mathbb{N}$ satisfy $n+m=m+n$".\\


An arbitrary element of $\mathbb{N}$ is most commonly denoted by $i,\ j,\ n$,or $m$, but any symbol, $a,\ b,\ c, \ldots$, can be used.\\

Several rules of arithmetic apply (some by definition, others can be derived) such as
\begin{eqnarray*}
ab&=&ba\\
a+b&=&b+a\\
a+bc&=&a+(bc)\\
a(b+c)&=&ab+ac\\
(a+b)+c&=&a+(b+c)\\
(ab)c&=&a(bc)
\end{eqnarray*}

Subtraction and division are not generally defined. In addition, we define one integer, $n$, to the power of another, $m$, to mean $n$ multiplied by itself $m$ times: $n^m=\underbrace{n\cdot n \cdot \ldots \cdot n}{m}$.

\begin{defn}
The power is an \textbf{operator} just like addition and multiplication, and is defined to have higher priority than the other two.
\end {defn}

\begin{xmpl}
If we have $x=4$ and $y= 2$ and want to evaluate $$x^y+y^x$$
then we replace the values of x and y in the expression, and evaluate it,
taking care to observe the correct order of operations:
$$4^2+2^4=16+16=32.$$
\end{xmpl}