base e

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base e

Postby philps » Fri Mar 13, 2009 6:14 pm UTC

Hello,

I am wondering how the design of a base e number system would look like. In base 3 there are the numbers 0, 1 and 2. In base 4 there are the numbers 0, 1, 2 and 3. I read somewhere that base e is the most compact way to store information, so that is why I am curious....

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Re: base e

Postby t0rajir0u » Fri Mar 13, 2009 6:25 pm UTC

It's possible to define base expansions with respect to any real number (you take the digits strictly less than the absolute value), but for non-integers you run into two major problems:

1) Sometimes base expansions aren't unique, and

2) Sometimes not all real numbers have a base expansion.

I'm not sure what you mean by "most compact way to store information," since for most people that prize usually goes to binary.
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Re: base e

Postby NathanielJ » Fri Mar 13, 2009 6:59 pm UTC

t0rajir0u wrote:I'm not sure what you mean by "most compact way to store information," since for most people that prize usually goes to binary.


He's likely referring to this.
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Re: base e

Postby t0rajir0u » Fri Mar 13, 2009 10:41 pm UTC

The formula used there doesn't really work at non-integer values; that was meant to be an approximation, not the answer. If you try to claim that base e is more efficient at representing information than either base 2 or 3, then I'd like to see you write 4 in this base. (You might guess that it has an infinite "decimal" expansion. You'd be right.)
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Re: base e

Postby gmalivuk » Sat Mar 14, 2009 12:26 pm UTC

t0rajir0u wrote:2) Sometimes not all real numbers have a base expansion.

Really? Can you give an example of this?
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Re: base e

Postby NathanielJ » Sat Mar 14, 2009 1:46 pm UTC

t0rajir0u wrote:The formula used there doesn't really work at non-integer values; that was meant to be an approximation, not the answer. If you try to claim that base e is more efficient at representing information than either base 2 or 3, then I'd like to see you write 4 in this base. (You might guess that it has an infinite "decimal" expansion. You'd be right.)


I agree that it doesn't work for non-integer values, but this is a mathematics forum and it's still sort of interesting (in my mind, anyways) that e would be the "best base" if we could somehow write a number using e different symbols. As it is though, 3 wins.

The argument that e isn't a "compact" base because 4 has an infinite expansion in it doesn't hold much water in my mind though. If you think either base 2 or 3 is more efficient at representing information than base e, then I'd like to see you write e2 in one of those bases.
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Re: base e

Postby jestingrabbit » Sat Mar 14, 2009 3:10 pm UTC

NathanielJ wrote:The argument that e isn't a "compact" base because 4 has an infinite expansion in it doesn't hold much water in my mind though. If you think either base 2 or 3 is more efficient at representing information than base e, then I'd like to see you write e2 in one of those bases.


Try this argument on for size then. Say you want to write down some reduction formulae that describe the base in a way that's purely symbolic. What I mean is, you can define the real numbers as decimal expansions but is it possible to do so for base e? Can you formulate the real numbers in terms of their e-expansions, without reference to anything but strings of symbols?

What you realise as you try to answer this question is that you'll need to determine a method for determining equivalence classes of strings of symbols. Now, perhaps you're much more cunning than I, but it seems quite difficult to write out some finite set of rules which tell you when and how a particular string can be rewritten, though in decimal that's pretty easy. A canonical form, which in decimal is merely "doesn't end in an infinite string of 9s" seems likewise quite intractable. Furthermore, doing practical additions and multiplications seems nigh impossible.

Now, this isn't to say that there aren't non integer bases that I like. I'm quite fond of base phi, but it has lots of positive points going for it that e doesn't have. Its quite easy to write down what the canonical form should be, and addition and multiplication are all about substituting strings of the form 011 with those of the form 100, and for convenience I also remember 0200 = 1001.

But there are no rules like that for e.
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Re: base e

Postby gorange » Sat Mar 14, 2009 8:31 pm UTC

Phil, this discussion of non-integer bases really peaked my interest; it wasn't something I had thought of before.

I googled 'base phi' and found this useful article on Phigits (great name).

From this:
There is a proof that any integer can be written as a sum of powers of this irrational number, phi. (Importantly, the sum does not need to repeat any power). Is there a proof that you can/can't do this with e?

t0rajir0u wrote:I'd like to see you write 4 in this base. (You might guess that it has an infinite "decimal" expansion. You'd be right.)

does your response about infinite expansion still apply in this respect?

jestingrabbit wrote:But there are no rules like that for e.

It seems like you are saying there does not exist a proof that says you can. Do you know of a proof that says you can't?

* Does anyone know why we define Phi = 1 + phi ?
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Re: base e

Postby antonfire » Sat Mar 14, 2009 8:43 pm UTC

If you could write an integer as a sum of powers of e (not all of them 0'th powers), then e would be algebraic. It isn't. The proof of this is not that easy, but the fact itself is well-known.
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Re: base e

Postby t0rajir0u » Sat Mar 14, 2009 10:56 pm UTC

gorange wrote:peaked my interest

Piqued.

gorange wrote:There is a proof that any integer can be written as a sum of powers of this irrational number, phi. (Importantly, the sum does not need to repeat any power). Is there a proof that you can/can't do this with e?

Every rational number requires an infinite number of "digits" to express in base e, and this representation is not, as far as I know, unique. And not in the way that base-phi representations aren't unique - there are no "local" shifts in digits that work, because any such relation would imply, as antonfire noted, that e is algebraic. That is, the non-uniqueness occurs at infinitely many digits. (For example, there is some "decimal" starting with 0.2... that sums to 1.)
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Re: base e

Postby gorange » Sat Mar 14, 2009 11:58 pm UTC

antonfire wrote:If you could write an integer as a sum of powers of e (not all of them 0'th powers), then e would be algebraic. It isn't. The proof of this is not that easy, but the fact itself is well-known.

Thanks, antonfire. Further searching has yielded: It is the Lindemann-Weierstrass theorem that shows that e and pi are transcendent, not algebraic.

What is interesting is that e being transcendent does not mean you can't get finite representations of rational numbers with e as a base-- it means that in order to do so, you would need irrational numbers as some or all of the coefficients of the polynomial. I've never seen an interpretation of a number system where the symbols were non-rational, but if such were done with base e, it would be possible to have finite sums of powers for some integers.

t0rajir0u wrote:
gorange wrote:peaked my interest

Piqued.

I think you have confused my remark with the phrase "piqued interest". Allow me to disambiguate: I really did mean that my interest had been maximized, not excited.
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Re: base e

Postby t0rajir0u » Sun Mar 15, 2009 12:21 am UTC

gorange wrote:it means that in order to do so, you would need irrational numbers as some or all of the coefficients of the polynomial. I've never seen an interpretation of a number system where the symbols were non-rational, but if such were done with base e, it would be possible to have finite sums of powers for some integers.

Yes, but if you allow irrational digits then you lose any hope of uniqueness. Furthermore, the digits you need would need to occur in finite extensions of \mathbb{Q}[e], which is a pretty arbitrary restriction, and in any case it is trivial to note that given any finite choice of digits at most countably many numbers have terminating expansions, so you're going to miss a cocountable set in either case - you might as well make choices that allow the countable set that do have terminating expansions to include the integers in a more-or-less natural way.

gorange wrote:I think you have confused my remark with the phrase "piqued interest". Allow me to disambiguate: I really did mean that my interest had been maximized, not excited.

Is it reasonable to assert that your interest in a subject is maximized at the beginning of your investigation of the topic? In my experience, this is rarely the case.
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Re: base e

Postby qinwamascot » Mon Mar 16, 2009 5:19 am UTC

t0rajir0u wrote:Every rational number requires an infinite number of "digits" to express in base e, and this representation is not, as far as I know, unique. And not in the way that base-phi representations aren't unique - there are no "local" shifts in digits that work, because any such relation would imply, as antonfire noted, that e is algebraic. That is, the non-uniqueness occurs at infinitely many digits. (For example, there is some "decimal" starting with 0.2... that sums to 1.)


Firstly, 0 can be expressed with finitely many characters, as can 1 and -1. But all other rationals this isn't true [/nitpick]

Second, are there countably infinitely many ways to write 1 in base e, or just finitely many? I lean towards countably infinitely many, which makes this seem much less useful for anything except expressing power series of e. I'd try to work it out, but my brain is fried right now.
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Re: base e

Postby jestingrabbit » Mon Mar 16, 2009 12:17 pm UTC

qinwamascot wrote:Second, are there countably infinitely many ways to write 1 in base e, or just finitely many? I lean towards countably infinitely many, which makes this seem much less useful for anything except expressing power series of e. I'd try to work it out, but my brain is fried right now.


I'd guess that there are actually an uncountable infinity, but I could easily be wrong. Finite make no sense though. Consider that as you can rewrite 1 in one way, then you can rewrite 0.1, 0.01 etc as well. So after you rewrite 1, rewrite another digit. Continue at will.
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Re: base e

Postby qinwamascot » Wed Mar 18, 2009 1:26 am UTC

That's what I was thinking as well. Which makes base e quite difficult to use. If it's not even trivial to check if two numeric expansions are equal, then clearly the system is not very efficient for normal computations.
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