Addition of natural numbers is defined inductively as follows: $$ n + 0 := n, \hspace{0.5cm} n + m^* := (n+m)^*$$

i.e. $$ \begin{aligned} n + 1 &= (n+0)^* = n^* \left( \text{the successor of } n \right), \\ n + 2 &= (n+1)* = (n^*)^* = n^{**}, \\ & \hspace{0.25cm} \vdots \\ n + m &= n^{* \ldots *} \left( \text{the } m^{th} \text{ successor of } n \right). \end{aligned}$$

Similarly, multiplication of natural numbers is defined inductively as follows: $$ n \cdot 0 := 0, \hspace{0.5cm} n \cdot m^* := (n \cdot m) + n$$

i.e. $$ \begin{aligned} n \cdot 1 &= (n \cdot 0) + n = n, \\ n \cdot 2 &= (n \cdot 1) + n = n + n, \\ & \hspace{0.25cm} \vdots \\ n \cdot m &= n + n + \ldots + n \text{ (m times)}. \end{aligned}$$

Associative laws

Addition and multiplication of natural numbers are associative, that is $$ \begin{aligned} n + (m + p) =& (n + m) + p, \\ n \cdot (m \cdot p) =& (n \cdot m) \cdot p. \end{aligned} $$

Commutative laws

Addition and multiplication of natural numbers are commutative, that is $$ \begin{aligned} n + m =& m + n, \\ n \cdot m =& m \cdot n. \end{aligned} $$

Distributive laws

Addition and multiplication of natural numbers are distributive, that is $$ \begin{aligned} (n + m) \cdot p =& (n \cdot p) + (m \cdot p), \\ p \cdot ( n + m) =& (p \cdot n) + (p \cdot m). \end{aligned} $$

Powers

Let \( a \) and \( m \) be two natural numbers then the \( m^{th} \) power of \( a \), \(a^m \), is defined inductively as follows: $$ a^0 := 1, \hspace{0.5cm} a^{m^*} := a^m \cdot a$$

i.e. $$ \begin{aligned} a^1 &= a^0 \cdot a = 1 \cdot a = a, \\ a^2 &= a^1 \cdot a = a \cdot a, \\ & \hspace{0.25cm} \vdots \\ a^m &= a \cdot a \cdot \ldots \cdot a \text{ (m times)}. \end{aligned}$$

Also \( a^m \cdot a^n = a^{m+n} \) and \( \left( a^m \right)^n = a^{m \cdot n} \).