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.
t0rajir0u wrote:2) Sometimes not all real numbers have a base expansion.
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.)
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.
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.)
jestingrabbit wrote:But there are no rules like that for e.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
gorange wrote:peaked my interest
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?
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.
t0rajir0u wrote:gorange wrote:peaked my interest
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.
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.
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.)
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.
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