Base is only a way of representing a number, a symbol for it. The numbers properties are separate from it's base. IE The number represented by 10 (in base 10) is divisible by the number resented by 5 (in base 10), in any base (or roman numerals or words or what ever symbols). I hope I explained that well enough.

And c is a rational number if and only if c = a/b where a and b are both integers. And pi cannot be written as a/b, so it is irrational in any base since base is irrelevant. Although a/b can be arbitrarily close to pi.

It's not rational in base pi, since 1 2 3 4 5 etc don't represent integers in base pi.

That is the definitation of rational in standard mathemtics.
[ulr]http://mathworld.wolfram.com/RationalNumber.html

As far as standard math is concerned the integers are [...-2,-1,0,1,2...] thats all of them (there is a concrete definition in standard mathematics). If you want to make up new "integers" then we'll not playing in standard mathematics. Now convert those to basepi and you'll have the integers in basepi, I think -2 to 2 go fine but then you get problems. You have to remmeber that operations on the numbers are independent of bases. IE if m basex = n basey. Then
m basex*m basex = n basey*n basey
m basex/m basex = n basey/n basey
m basex+m basex = n basey+n basey
m basex-m basex = n basey-n basey

For fun.

Using my best guess about how a irrational base would work (I thought about it a bit, it gets really ugly)

0 1 2 3 are representable in base pi.

0basepi=0base10
1basepi=1base10
2basepi=2base10
3basepi=3base10
10basepi=pibase10
11basepi=pi+1base10
321basepi=3*pi^2+2pi+1 base10

There all kinds of wierd things with this 2basepi+2basepi=2base10+2base10=4base10=?basepi.
4basepi is as meaningless as 3base2 I think it's about 10.2201....basepi. So 2basepi+2basepi=10.2201....basepi. If you want to go with multiplication 2basepi*2basepi=10.2201....basepi.

at the end of the day 10 is just a base we picked, and whether we use base 11,121,12321 ...
3base76 = 3base21
its just the symbols(representation) we use to mean the different numbers;

therefore the basic mathematical operations still stay the same.
using base pi would be weird but the numbers will still stay the same
e.g
1 , 2 , 3 ... 10.858407346

i don't think some of you get a few subtle details still ...

the only "normal" bases, meaning bases where the standard assumptions of positional mathmatics ALL hold true are the natural number bases greater than 1 ... think about it for a minutes ...

Natural numbers = 1, 2, ...
Whole numbers = 0, 1, 2, ...
Integers = ..., -2, -1, 0, 1, 2, ...

for ALL natural number bases (and only those)

this is the system of positional representation

11.1 = 1 * base ^ 1 + 1 * base ^ 0 + 1 * base ^ -1

so this number, 11.1 means different things in different bases, but for all whole number bases you can do all normal math operations upon it, including representing all integers, etc ...

now how many symbols must you have to use a base ... the same number as the value of the base ... base 3 requires 3 symbols, base 10 uses 10 symbols, base 16 uses 16 symbols ...

and doing addition you say stuff like:

123
+ 42

if this is in base 5 ...
3 + 2 == 5, so carry a 1, remainder 0
1+2+4 == 7, so carry a 1, remainder 2
1+1 == 2, so carry nothing, remainder 2

so 123 + 42 = 220 in base 5 ...

now imagine a pi based number system ... which sounds perfectly reasonable and logical at first ...

how many symbols do we need?

3.14159...

well, already I see a problem, but lets try to just ignore it for a minute (I'll use 4 symbols for now, representing 0,1,2,3 and ignoring pi ..., just like base 4 has 0,1,2,3 and no 4).

so we have a number, 123. What does this number mean (in decimal terms)?

1 * pi^2, + 2 * pi^1 + 3 * pi ^0

simplified

pi^2 + 2pi + 3

so far, not great, but not obviously broken ...

now try to ADD, 123 + 22 (22 == 2pi + 2)

3 + 2 == 5, so how many pis come out of 5 (1) and leave how many 1s? I can't do it ... can you?
2 + 2 == 2, so how many pi^2 come out of that (1) and leave how many pis? I still can't do it ...

so you can't ADD in base pi ... because the ONES digit is not 1 pi ... it's ONE ... the real one, as in "If there is only 1 creature of a kind, that creature is unique" ... the number 1 has the same meaning in EVERY positional base that makes since ...

But to invent a thing LIKE a base (but not quite the same) for uing pi, you would have to change the fundamental rules ... you would have to say 1 == 1 pi, not normal 1 ... and in that system, what does 12.1 mean? does it mean 1*pi^2 + 2*pi^1 + 1*pi^-1?

That might be usefull in many circles, but it is definately exhibits different behaviors than the whole number bases ...

just as many ancient cultures had no formal 0, and had many hard times expressing certain important ideals, can you imagine trying to use a number system in which the concept of 1 is incompatible with your symbols?

hope this post at least makes you think :)

Like people have said before, rationality has nothing to do with the base that we represent numbers by. We just often think about the difference between them and irrationals as repeating decimals and nonrepeating.

The reason why base does not matter in the second way of thinking about it is the following. If there exists a repeating sequence after the decimal point in any base, we can actually construct a fraction which that number represents, and so it's rational. Conversely, if a number is rational, the expression in that base is forced to repeat. Hence, the two ways are equivalent ways of thinking about rational vs irrational, and are independent of base.

You can express a number x in a base b like this:
1) find the largest power of b that is smaller or equal than x (say it's b^l)
2) if l<0, l=0
3) output floor(x/b^l)
4) decrease x by b^l*floor(x/b^l)
5) decrease l by 1
6) if l==-1, output `.'
7) go to 3

This procedure doesn't need an integer base. Actually, it works just fine with an irrational base, and we could use it to define what it means to use a base that is an arbitrary real number b>1. The natural numbers up to 10 in base pi are:
0
1
2
3
10.220122021121110301...
11.220122021121110301...
12.220122021121110301...
20.202112002100000030...
21.202112002100000030...
22.202112002100000030...
100.01022122221121122...

This is not a very interesting base, since operations are hard to make.

A more interesting base is phi:=(1+sqrt(5))/2. Google for "base phi" and you'll see that it actually has applications. Base phi only uses 0 and 1, two 1s never appear next to each other, and the integers are the numbers that have symmetrical representations in base phi (the digit of phi^k is the same as the digit for phi^-k, for all k).

i would have thought that if you had base pi you would go up in integer counts until you reached pi:
1 , 2 , 3 , 10^1 + (pi-3) , [10^1 + (pi-3) + 1] , ...
why would it not work like this?

Edit: changed '-' to ':'

I was playing around on my calculater in math class today and i discovered something quite intersting ( i'm sure it's been discovered before but... )

e^( pi * i ) = -1

yay! you can represent -1 with only symbols =)