A long standing tradition in Science Faction states that a species can show that it is inteligent by demostrating prime numbers.
but what if they don’t use a base ten mathmatical system?
is 13 still prime in base 8? in hexidecimal?
Charles
Yes, of course!
In base 8, “15” is prime, and in base 16, “D” is prime, but these are all ways of saying that 13 (in base 10) is prime.
It just so happens that in bases 8 and 16, “13” is equal to 11 and 19, respectively, so in this case, it works out. But, for instance, “13” in base 12 means 15 in base 10, so it won’t be true that the symbol 1 followed by the symbol 3 will always denote a prime number.
If we were to find some sort of alien document with mathematical symbols on it, we would need to determine how they count. But this shouldn’t be too hard, if the document is long enough. If they use a number system like ours, just count the number of distinct digits. In our case, there are 10 (0123456789), so we know we use base 10.
well it’s still 13 isn’t it? even if it is 8 + 5. If you ignore the nomenclature for a moment and just put 13 objects on the ground then try and divide them into equal whole portions, you can’t, regardless of the counting system you use.
The confusion here is between a number and its representation. Primality is a property of a number, not its representation, and is therefore independent of any particular representation.
I have another question… Would a number be prime in say, base .25?
You can’t have fractional bases (at least not in any number system I know of). You need at least one symbol to count things.
In Contact, the aliens sent their prime numbers as a series of pulses. Base 1. That was pretty easy to figure out. 
Why can’t you?
Let’s see… Base 1/2…
(1/2)^0 is 1, like normal.
(1/2)^1 is 1/2…
(1/2)^2 is 1/4…
So far you can’t get any higher than almost two, but you COULD use the decimal point, where the numbers would bet progressively bigger.
Of course, in this case it’d be trivial, since it’s exactly like a reverse binary. I’ll quantify my question into something more interesting, like base e, or some other slightly less rational number.
Interesting hijack, regarding non-integral bases. In base “e”, would e be “rational”, and all the current integers become transcendental? Would ‘pi’ still be transcendental in base “e”? Ouch, my brain hurts.
No, DarrenS, if I understand it, all numbers would retain their rationality and transcendtiality. The set of Integers would remain the same no matter the base representation, and the set of Rationals and the set of Algebraics are both defined in terms of the set of Integers.
The question could also be defined as “is one always one no matter how you count?” AFAIK, the answer is yes. I haven’t heard a good argument to the contrary yet (not that one couldn’t necessarily be made).
I guess it depends on if there is any x such that x^0 != 1 and x could be used as the base for a number system.
I don’t even know if that would cut it. One is not defined as “x^0, where x is your base”. Er, well, is it? I don’t even know how one is defined!
Of course, 0^0 != 1, but for that very reason, 0 can’t be used as the base.
Rationality and transcendentality are also properties of a number, not its representation, and so they stay the same no matter what base you’re in. Fractional bases were covered in detail here, and this thread may be of interest as well.
IMHO, the main thing that prevents bases other than those based on positive integers from being useful is the difficulty of figuring out division. Addition, subtraction, and multiplication are all pretty straightforward, but division is tough.
friedo: You certainly can have fractional bases. They’re fractals.
No.
Writing a number in base 8 no more affects its primality than writing it in Roman numerals would. What (if anything) a number is divisible by is a property of the number itself, while bases have to do with how the number is written.
This thread makes my brain hurt.
I do know this, however; if I count out 13 Oreo cookies, I cannot divide them into equal groups of whole cookies other than one group of thirteen cookies or thirteen groups of one cookie. It does not matter what words I use to count them or what numbers I visualize in my head as I count them; if I have this many cookies
O O O O O O O O O O O O O
I can’t subdivide them evenly in any way except as singles. And that’s what a prime number is, and that’s the only way I understand it.
Now I am hungry.
I am now slapping myself upside the head. I got base confused with exponent.
Thanks, ultra.
No problem.