Recall that a number in base is an expression
where the are nonnegative integers less than . So, for instance, the number is represented as in base .
Although most applications of bases restrict to be a positive integer, it is worth considering what happens when is, more generally, a positive real number.
Let be a real number. The -expansion of a nonnegative real number is a representation
where the are nonnegative integers less than .
Note that is required because there need to be some positive integers less than (otherwise there are no nonzero choices for the ).
From this definition, it is straightforward to deduce the following fact:
For any , every nonnegative real number has a -expansion.
The algorithm to generate a -expansion of a real number is the familiar one: first let be the integer satisfying , and then let Then and , so the process can be iterated: let and let and so on.
Let be the golden ratio . Find a -expansion of .
First, , so the expansion starts . Now , so the expansion starts . Since , this will need to be expressed using negative coefficients. In fact, so , so the -expansion of is . In base notation, this is
Note that this representation is not unique: for example, since , it is also true that
This illustrates a situation that is different for non-integer bases: finite (terminating) integer-base expansions are always unique but integer-base expansions are not unique, e.g. in base , as discussed here However, finite base- expansions may not be unique for algebraic integers .
The golden ratio is the larger positive root to .
We can calculate that
This allows us to write numbers in base , where each place value is a non-negative integer less than . For example,
Give the finite decimal representation of 5 in base that doesn't use a consecutive pair of 1's.
(You may assume the fact that a finite decimal representation with no consecutive pair of 1's is unique. This is known as the standard form for base .)