Why the quotient of the circumference over the diameter is an irrational number? In the decimal system, circumference / diameter = 3.14159265359...
But, for example, in binary, that ratio is 11.00100100001111110111...
Or in hexadecimal, 3.243F6A8885A308D313198A2E037073...
We also obtain an irrational number, independent of the numeric system that we use.
Has something to do with the fact that we use decimal system, or with the dimensions of the given space?
Could it be otherwise, with another set of rules or number of spatial dimensions? (Obtain an exact number)
 A: The representation is irrelevant since the repeating pattern forces the number to be rational. Say we have a number that repeats eventually in base $n$ called $r$ and say the digits repeat after $k$ digits.  In general there will be an integer portion, a non-repeating portion after the decimal, and then the repeating portion. The first step is to find some appropriate power of $n$ so that the repeating pattern starts at the decimal point. So say I have $r=0.124737373...$ in base $8$ I would multiply by $8^3$ so that $8^3s=124.737373....$ and from here we then multiply by $n^k$ to move the pattern to match up with itself again. So in our running example we would have that $8^28^3s=8^5s = 12473.737373...$ and now we subtract to remove the portion after the decimal place so in our example we take $8^5s -8^3s = 12473.737373... - 124.737373... = 12349$ and so we have that $(8^5-8^3)s = 12349$ Simply noticing that $8 = 10$ base $8$ this gives us $(10000-1000)s = 1700s=12349$ and so solving for $s$ we recover $s=12349/1700$. This works in every base just like it does in base 10. It's a property of the notation itself rather than the base.
