This is probably either a deep thought or patently silly, but here it goes:
Is there a different number system, say base 3 or base 697, where the value of pi would be a whole number?
Yep. Base pi. It’d be 1.
No, base pi it would be 10. But it makes little sense to use a non-integer as a base.
Assuming that we’re talking about integral number bases, and not trying to use an irrational number as the base:
No.
Repeating decimal fractions, (like 0.333333 for 1/3 or 0.714285714285… for 5/7) are often expressed as terminating decimal fractions in some other number system. 1/5, 0.2 in decimal, is endlessly repeating in binary. 1/3 and 2/3 are 0.4 and 0.8 respectively in base 12
I’m pretty certain that a non-repeating decimal fraction, such as the square root of two, ‘e’, or pi, is always a non-repeating decimal no matter what integral base you’re using. I’m not sure of a proof for this, (and if there is one it’s probably very complicated,) but the fact that these expressions form a completely different class of number from the fixed and repeating fractions, irrational numbers versus rational numbers, backs me up I think. There’s nothing else in the various number classes that varies based on what base system you’re looking through, after all - this is fairly fundamental stuff.
I wouldn’t call the question ‘patently silly’, but it doesn’t really lead to any great new understanding of pi as far as I know.
To follow up on Ethilrist’s answer, you could also use a base of pi/3, pi/8 square root of pi or whatever you choose, but all of these are also irrational numbers, and probably trancendental. (There’s an outside chance that pi divided by eight is one over the square root of seven or what have you, but that’s not very likely at all.)
<- math geek and proud of it!!
In part it depends on what you mean by “whole number”. If by “whole number” you mean “integer”, then there is no base in which pi is a whole number–being a whole number is a property that is independent of what base you are in.
On the other hand, if by “whole number” you mean a number which can be represented in a given base as “ab…cd.00000000…” (i.e., only zeros after the “decimal” (radix) point), then, yes, there are bases in which pi is a whole number–base pi, for example, as was previously mentioned, in which pi is represented as 10.
That’s correct, and not hard to prove. If a number does have a repeating expansion in an integral base, it’s a geometric series (e.g, .333… in decimal is 3/10 + 3/100 + 3/1000 + …). Any geometric series with rational terms converges to a rational number. So a number with a repeating expansion (in any integral base) is a rational number. Contrapositively, an irrational number does not have a repeating expansion in any integral base.
There’s no chance that pi/8 is not transcendental. Transcendental means that it is not the root of any polynomial with integral coefficients. If you can find a such a polynomial with pi/8 as a root, then I can find a polynomial with pi as a root, contradicting pi’s transendence. In fact, any finite sequence of “algebraic” operations you can perform on pi (such as integral exponentiation and root taking, adding a rational to it, subtracting a rational from it, dividing by a (nonzero) rational, or multiplying by a (nonzero) rational) will give you a transcendental number. So, for example,
pi + 7
pi - 3
5pi
pi/8
pi[sup]27[/sup]
pi[sup]1/391[/sup]
pi[sup]27/391[/sup]
are all transcendental.
The proof that an irrational number is represented by a nonrepeating decimal expansion is not dependant on the base used for the expansion (so long as the base is an integer, and non-integer bases aren’t well defined anyway). Therefore, if something is non-repeating in any integral base, it’s non-repeating in all integral bases.
And there is not an outside chance that Pi/8, or any other simple arithmetic combination of Pi and algebraic numbers, is algebraic. If it were, then there would exist a rational-coefficient polynomial to which that number would be a root, and from that polynomial one could easily construct another polynomial with Pi as a root. But that’s impossible, because Pi is trancendental.
0 * pi 
You can define non-integer bases pretty easily. They’re not terribly different from integer bases except that you can’t use repeating/non-repeating representations to differentiate between rational and irrational numbers.
Whoops, my mistake, I’m a little fuzzy on transcendentals I guess.
So, is it in COMBINING transcendentals that there’s a one-in-a-billion chance that the result is not transcendental? for instance (4*pi) + (e/2)? I’m pretty sure I remember something like that in an old Martin Gardner article.
by the way, my use of one-in-a-billion there was not meant to be a mathematically sound probability, more like ‘one chance in an extremely extremely large number that we’re not sure how large it is yet.’
pi and 1 - pi are both transcendental, but pi + 1 - pi is algebraic. There’s a whole class of pairs like that (one for every transcendental/algebraic pair).
Yeah, that’s right. As far as I know, it is not known whether numbers such as e + pi, e * pi, e[sup]pi[/sup], pi[sup]e[/sup], or ln(pi) are transcendental or not.
In fact, as far as I know, some of those listed above may even be rational! It’s an interesting fact that an irrational number raised to an irrational power can produce a rational number. I’ve always thought the following was a rather clever proof of that:
sqrt(2) is known to be irrational.
sqrt(2)[sup]sqrt(2)[/sup] is either irrational or rational. If it’s rational, we’ve demonstrated that irrational^irrational can be rational. Otherwise:
sqrt(2)[sup]sqrt(2)[/sup] is irrational and (sqrt(2)[sup]sqrt(2)[/sup])[sup]sqrt(2)[/sup] = 2 (obviously rational), and we’re done!
Along those same lines, I still get a kick out of the fact that i[sup]i[/sup] is a real number. But the proof is a little more involved than that clever sqrt(2) one.
And chrisk, you seriously underestimate the probability there. There are an uncountable number of trancendental numbers, but only a countable number of algebraic numbers, so the chances are literally infinitesimal that any given random combination or trancendental numbers will result in an algebraic number. There are, of course, combinations which will work, as illustrated by ultrafilter, but then also, ultrafilter wasn’t choosing his combinations randomly.
That surprised me too. Is i[sup]i[/sup] (which equals e[sup]-pi/2[/sup], plus or minus some branche cuts I don’t want to deal with) transcendental? My intuition says it’s not, but I’m not up to proving it.
In addition to using non-integers as bases, one can also use variable radix bases. I.e., least sig. digit is base 10, the next base 2, etc. using some predefined pattern for the radix of each position. I can assume that John Conway knows of such a radix system in which Pi is looks quite nice. (He knows a lot of strange and wonderful formulas involving Pi.)
If you use the L/R matrix system (from the Stern-Brocot tree) for representing reals, e has an extremely nice pattern, but it’s not strictly repeating like a rational number.
Fixed radix number systems are so 20th Century…
Can somebody please explain to me how to use non-integer bases?! It’s my understanding that a number base simply defines what you are taking the power of when you move from position to position in an integer.
10 base pi would be pi base 10. OK, so we start off counting 1, 2, 3, now what comes next? 10 comes between 3 and the next integer!
Actually, there’s no need to discuss it from the point of view of a transcendental base. Please tell me how I count how many pennies I have when I am using base 3.5?
um, if pi/8 satisfies a polynomial, so does pi…
As are numbers in general. I don’t know a mathematician who really does anything with them.
I don’t know much about such things, but I’ll have a stab at it.
Considering that 10 is pi, and 1 is one, it would seem to follow that 0.1 is 1/pi and 0.01 is 1/(pi^2).
Now if you’re counting one, two, three, four, then four lies somewhere between 10 (3.14… in decimal) and 11 (4.14… in decimal). 1/pi happens to be about 0.318, so 10.1 base pi is about 3.142 + 0.318 = 3.460, considerably less than 4.
So now let’s try 10.2, which is pi + 2(1/pi), or about 3.778.
10.3 turns out to be about 4.097, and if you needed more accuracy, you would use more “digits” (pi-its?) just like we use 3.14, 3.14159, or 3.1415926535897932384 for pi, depending on how much accuracy we need. I suppose it would be easy to prove that you’ll never represent 4 with 100% accuracy in base pi, just as you’ll never represent pi with 100% accuracy in base 4, or any other integral base.
Now, you want to try base 3.5? Okay… 1, 2, and 3 are trivial. I get 10.1220… for 4, where 10 is 3.5, 0.1 is 1/3.5=2/7, 0.01 is 1/3.5^2=4/49, etc.
1(3.5) + 0(1) + 1(1/3.5) + 2(1/3.5^2) + 2(1/3.5^3) + 0(1/3.5^4) = 3.99562682
I don’t know if the “3.5-imal” form of 4 terminates, repeats, or goes on like an irrational does in integral bases, but I’m sure someone around here knows, or can figure it out.
It’s not particularly easy, and I don’t think anyone ever claimed it was useful or convenient. I think it would be fun to have bases less than 1. If I’m not mistaken, 34.6 base ten would be 64.3 base one-tenth.
I can suggest at least one number representation which is very common and uses both a non-integer base and varing radices: a digital 24-hour clock.
If you have a clock displaying time as hhmmss, you can look at it two ways:
(1) hh is base 24, mm is base 60 and ss is base 60 (using two decimal digits to display each digit in the fraction of a day)
or
(2) The first h is base 2.4, the first m is base 6 and the first s is also base 6, while the other digits are all base 10. So the first digit is a fractional base, and the full representation has three different radices.
You could also view a 12-hour clock in the same way, with the first digit base 1.2, and the am/pm part a way of representing a binary digit, i.e. a digit using base 2 arithmetic – i.e., 4 different radices (1.2, 2, 6 and 10) in one number representation.