A number base is a way of writing numbers. We use base . A number in base we can write a number
Where , , . It is possible to define bases with non-integer base but for now we only consider integers greater than .
Properties
Theorem
In a base any given integer has digits.
Proof:
Let have digits, therefore . Since is increasing, to obtain the max we require maximal therefore . For a minimum of we require . So we have .
Theorem
A representation of base is unique.
Proof:
Assume for the sake of contradiction that there exists two distinct representations of base . Therefore and . First we claim that .
Next consider . We get . Consider it . We get . And since . Then consider then and so on and we find that every digit is identical therefore we get a contradiction so the representation is unique.