Integers

The integers can be thought of as the natural numbers extended with an element \((-1)\) such that \((-1)^2=1\) so \(\mathbb{Z} = \mathbb{N}(a)/\langle a^2=1\rangle\). Or with an equivalence relation \(\mathbb{Z} = \mathbb{N}^2/\langle (x,y)\sim(w,z) \iff x+z=w+y \rangle \). They can be defined by a pair of natural numbers. To create a positive integers and negative integers as: For convenience we can also just write the number as a decimal and it will convert it to a natural number. We can multply two natural numbers together: Equality:

Internal Representation

For speed and efficiency, the internal representation of natural numbers is not S(S(S(S(...0)))) but uses machine arithmetic. $$\sqrt{1}$$