You're probably familiar with the different integer bases: for example, is , or . Why? Because . This should be old news to you. If you need a refresher on bases and base conversion, read through this note: Number Base Conversion
But do bases have to be integers? For example, is there any meaning to something like base ""?
Well, let's try it. Any number
So what would be in base ?
It would be . Interesting.
But how would we convert to base ? Well,
Everything more or less is going as expected right now. We have that , and .
But things in aren't always normal. Let's compare the numbers with :
Wait, what? We should have , because that's just common sense. But clearly . Where did we go wrong?
Rest assured, nothing went wrong. Base is just a peculiar base that has these sort of strange properties.
A question is then brought into mind: For what bases does intuition fail us, that a digit numeral is larger than a digit numeral? Think about it yourself before reading below.
To solve this problem, we compare the largest digit numeral with the smallest digit numeral in base . First off, we know that because when , the normal intuitive results hold (it's just binary). Also, or else there cannot be any digit except . Thus, the smallest digit numeral is
and the largest digit numeral is clearly
Thus we want
When , which means , then which is not possible.
When , which means , then it turns out where is the golden ratio. This brings up a good point: because of the identity .
When , which means , then where the upper limit is the only real constant that satisfies the identity .
As increases, the upper bound tends to . Thus, for sufficiently large , all bases under and greater than exhibit the strange property of a digit numeral being larger than a digit numeral.
I hope you found this little tidbit of non-integral bases and their strange behavior pretty interesting. Thanks for reading!
As an exercise, see if you can:
somehow relate base to base
prove or disprove that every integer and real number of the form can be uniquely represented in base .