# Rational Bases

I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do this:

The problem is: Write 5 in base 3/2

5*(2/3)= 3 remainder 1
6*(2/3)= 2 remainder 0
2*(2/3)= 1 remainder 1
1*(2/3)= 0 remainder 2

Take 2101 and divide each digit by 2.

5 in base 3/2 = 2/2 1/2 0/2 1/2

Check:

(2/2)*(3/2)^3+(1/2)*(3/2)^2+(0/2)*(3/2)^1+(1/2)*(3/2)^0=
27/8+9/8+0+1/2= 5

It worked.

Is it okay to extend digits to rational digits?
Is this a valid way to put integers in rational bases?

I have worked out a way to put rational numbers into rational bases but the process is complicated.

CRGreathouse
Homework Helper
Is it okay to extend digits to rational digits?
Is this a valid way to put integers in rational bases?

Yes to both. You can even use irrational bases; phi is a popular example.

Thank you CRGreathouse,

Onward to irrational and complex bases!!

CRGreathouse
Homework Helper
Thank you CRGreathouse,

Onward to irrational and complex bases!!

For complex bases, Google "quarter-imaginary base".

You can read "The Art of Computer Programming" by Donald Knuth, vol I. A great (I don't mean sort of nice, I mean *great*!) book for anyone who has enough brain to ask a question like that. It is readable and humorous as well as seriously mind-stretching. Knuth doesn't need to be obscure to be impressive.
Have a go at negative bases. They are really entertaining.
Here is another, more important one: consider compare examples like base ten, hundred, and thousand. Also 2, 4, 8, 16 etc.
Then, for humorous effect, try root ten, root two, etc.
Try interconversions and arithmetic in each base.
It is a tremendous field for entertaining exploration.

Cheers,

Jon