Base pi

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Base pi

Postby webb.am » Wed Feb 24, 2010 7:08 pm UTC

Does anyone else find non-integer bases fascinating?

10π = π
A circle with diameter 1π has a circumference of 10π

Yeah... I'm not a mathematician so I can't really say anything interesting about them, but what about base π, base e, base φ? What does e look like in base π? Are non-integer bases interesting or am I just making something out of nothing?

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Re: Base pi

Postby the tree » Wed Feb 24, 2010 9:28 pm UTC

webb.am wrote: What does e look like in base π?
Maybe equally, if [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent, one can't really be expressed in terms of the other. But if they are algebraically dependant then it should be a finite expression. Whether or not [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent is an unsolved question.

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Re: Base pi

Postby GoldenPhi » Wed Feb 24, 2010 11:24 pm UTC

Interestingly enough, integers in base φ are finite decimals. For example 2 is 10.01 in phinary. Also since φ² = φ +1 any number can be represented without a 1 appearing next to any other 1 in its decimal form: 3 = 11.01 or 100.01 base phi.

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Re: Base pi

Postby skeptical scientist » Wed Feb 24, 2010 11:50 pm UTC

Mostly I don't like non-integer bases as you give up unique representation. Base phi is a special case, because there is a trick that gives a unique representation (as noted above). So base phi is really quite nice. But I'm not sure if it's useful for anything.
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Re: Base pi

Postby phlip » Thu Feb 25, 2010 12:13 am UTC

$$2.2021201002111122\cdots_{\pi} = e$$
Not sure what that achieves, though.

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Re: Base pi

Postby skeptical scientist » Thu Feb 25, 2010 5:24 am UTC

phlip wrote:Not sure what that achieves, though.

A very strange use of \cdots.

Wait, you can use double-dollar signs for display math? $$I \, did \, not \, know \, that.$$ Unfortunately, single dollar signs don't seem to $work$.
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Re: Base pi

Postby phlip » Thu Feb 25, 2010 7:36 am UTC

skeptical scientist wrote:A very strange use of \cdots.

Well, what's the norm for ellipses in tex? I thought cdots was good for "and so on". Is there a different one for when it's a number, and not, like, a matrix or something?

skeptical scientist wrote:Wait, you can use double-dollar signs for display math? $$I \, did \, not \, know \, that.$$ Unfortunately, single dollar signs don't seem to $work$.

Yeah, I stumbled onto it by accident when I was looking through some of my old posts and I saw that \( and \) got converted in a post, and looked at the help file, which says that \(this\), \[this\] and $$this$$ are supported. $this$ is supported, but turned off by default, so that it doesn't mess up normal uses of the dollar sign.

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Re: Base pi

Postby skeptical scientist » Thu Feb 25, 2010 7:56 am UTC

phlip wrote:
skeptical scientist wrote:A very strange use of \cdots.

Well, what's the norm for ellipses in tex? I thought cdots was good for "and so on". Is there a different one for when it's a number, and not, like, a matrix or something?

You know about both \cdots and \ldots, right? Generally you use \ldots for decimal expansions and sequences, and \cdots for things like sums and products. I'm not sure what exactly the rules are, but I think $$\pi=3.14159\cdots$$ is less natural than $$e=2.7182818\ldots$$...if you'll pardon the pun.
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Re: Base pi

Postby phlip » Thu Feb 25, 2010 7:59 am UTC

So noted. I've never really learned anything TeX-related properly... just picked it up from osmosis.

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Re: Base pi

Postby Talith » Fri Feb 26, 2010 11:42 am UTC

You can use \dots instead of \ldots which I think implies that dots on the line of writing are more commonly used than their centered colleagues. $$\dots \mbox{} \ldots$$

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Re: Base pi

Postby skeptical scientist » Fri Feb 26, 2010 5:39 pm UTC

Talith wrote:You can use \dots instead of \ldots which I think implies that dots on the line of writing are more commonly used than their centered colleagues. $$\dots \mbox{} \ldots$$

Actually, if this were real Latex, I think \dots is implemented in such a way that it automatically chooses between centered and lowered dots based on context. However, it seems that jsMath doesn't do this, and always uses lowered dots for the \dots command. So I don't think your conclusion is justified.
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Re: Base pi

Postby Talith » Fri Feb 26, 2010 7:01 pm UTC

I think we've gone slightly off topic, but thanks for disproving my hypothesis.

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Re: Base pi

Postby skeptical scientist » Fri Feb 26, 2010 11:47 pm UTC

Talith wrote:I think we've gone slightly off topic, but thanks for disproving my hypothesis.

Yes and no. The topic was "are non-integer bases interesting?" The fact that the thread went off topic is very relevant, and tells as that the answer is no. :P
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Re: Base pi

Postby Talith » Sat Feb 27, 2010 1:35 am UTC

Give it some credit, I see 5 posts on topic. That's mildly intriguing at the least!

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Re: Base pi

Postby jestingrabbit » Sat Feb 27, 2010 9:42 am UTC

Probably the most interesting thing about non integer bases is that its hard to work out what the admissability criteria for the representation of a number should be. For instance, in base [imath]\sqrt{2}[/imath] its pretty easy to see that there are two representations of 2: 100 and something very messy. That's not true of integer bases (just never let a representation have a number value that exceeds the base, and don't let it end in an infinite string of bbbbbb...), nor in fact phinary (never let 11 appear and never let it end with 10101010101...). But in more complicated bases, its less clear what we can and shouldn't allow.
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Re: Base pi

Postby phlip » Sat Feb 27, 2010 11:11 am UTC

Maybe a rule like... Take the representation at any point, cut off everything before that point, and move the decimal point (or whatever you call it in an arbitrary base) to just before the cutoff. So, for instance, you could take abcde.fghijk... and get 0.cdefghijk... or 0.ijk... or many other things. If the resulting number is >= 1, then the representation is bad. If the resulting number is <1 for all possible cutoffs, the representation is good.

So, for integer bases, this means no 0.999...-like representations, because 0.999... = 1.
For phinary, 0.11abcd... >= 0.10101010... = 1, so we don't use any representations that contain 11 or end 101010....

Equivalently, use the representation that comes last in lexical order (when they're prepended with 0s to the same length)... if you can increase the bn place and decrease the bn-1 and lower places (without making them go negative), and still end up with the same number, then do so.

So 2 is 100√2 and not 10.01000001001...√2, because the latter representation should "carry" upward.

This rule is a lot harder to judge for weird bases than it is for integers, though... would you know that 10.01000001001...√2 was a non-standard representation if it wasn't pointed out?

As a bonus, this also generalises the fact that all digits need to be in [imath]0 \le d < \left\lceil b \right\rceil[/imath]... you just have to say that the digits must be non-negative. Because if you try to have a digit d > b, then it'll be nonstandard, since 0.d > 1.

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Re: Base pi

Postby skeptical scientist » Sat Feb 27, 2010 5:25 pm UTC

Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.
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Re: Base pi

Postby jestingrabbit » Sat Feb 27, 2010 7:01 pm UTC

skeptical scientist wrote:Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.


Yeah, that works fine for determining the representation of a number given its value, but if we start with a representation, like we might have after we do an addition like (21/2) + (2 - 21/2), I think its a lot harder to get to the representation that we want.

The good thing about integer (and other nice) bases, is that we can calculate with them, but there are others where we can't really calculate with them in an easy way.
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Re: Base pi

Postby tmim » Sun May 16, 2010 11:38 pm UTC

What about the [imath]0.999\ldots[/imath] like representation of pi in base pi?

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Re: Base pi

Postby nash1429 » Mon May 17, 2010 7:02 am UTC

If we are including rational bases in this discussion I would be tempted to put things in bases less than 1 to mess with peoples' minds.

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Re: Base pi

Postby skeptical scientist » Mon May 17, 2010 8:15 am UTC

nash1429 wrote:If we are including rational bases in this discussion I would be tempted to put things in bases less than 1 to mess with peoples' minds.

I'd just read it backwards as a number in base 1/b.
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Re: Base pi

Postby gmalivuk » Mon May 17, 2010 3:19 pm UTC

tmim wrote:What about the [imath]0.999\ldots[/imath] like representation of pi in base pi?
I'm not sure what you mean. But one thing to note is that there's actually a whole interval of things that can have multiple representations, instead of just certain points as in integer bases. For example, while 9.999999...=10 in base-ten, 3.33333... > 4 in base-pi.
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Re: Base pi

Postby BlackSails » Mon May 17, 2010 4:21 pm UTC

By far the best base is base i*pi.

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Re: Base pi

Postby levantis » Wed May 19, 2010 10:04 pm UTC

i`d rather say it`s $$\sqrt{10}i$$, because you don`t need the usual base-changing algorithm to see how much is 123.45 (it`s $$-97.5+1.6\sqrt{10}i$$)

athough technicaly it is an integer base, i also like something like -2 or -10.

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Re: Base pi

Postby Mavrisa » Wed May 19, 2010 11:39 pm UTC

I like quater-imaginary...
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Re: Base pi

Postby phlip » Thu May 20, 2010 12:49 am UTC

levantis wrote:i`d rather say it`s $$\sqrt{10}i$$, because you don`t need the usual base-changing algorithm to see how much is 123.45 (it`s $$-97.5+1.6\sqrt{10}i$$)

I get just [imath]-7.5 + 1.6\sqrt{10}i[/imath]

But yes, to write a given complex number x+iy in base bi (for real x,y,b), you just encode x and y/b in base -b2 and interleave the digits of the two (lining up the decimal-or-whatever-they're-called points, and putting the imaginary part to the left of the real part for a given digit). So -7.5 and 1.6 in base -10 are 13.5 and 2.4... interleaving them gives 123.45.

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Re: Base pi

Postby Eastwinn » Thu May 20, 2010 11:04 pm UTC

the tree wrote:
webb.am wrote: What does e look like in base π?
Maybe equally, if [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent, one can't really be expressed in terms of the other. But if they are algebraically dependant then it should be a finite expression. Whether or not [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent is an unsolved question.


You can write [imath]e[/imath] in terms of [imath]\pi[/imath]... [imath]e = (-1)^{1 \over i\pi}[/imath] ... but that really doesn't get you anywhere at all. :P
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Re: Base pi

Postby black_hat_guy » Wed Aug 04, 2010 4:10 am UTC

e
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Re: Base pi

Postby Xanthir » Wed Aug 04, 2010 5:27 am UTC

Nah, i is a crappy base by itself, because it's basically complex-unary. Look at how, to make 20+17i, you had to write it as 9+9+2 + 9i+8i.

(Pretty sure you did that wrong, though - remember that the first digit is worth i^0 = 1, not i. So it should be 200890099.)

Quater-imaginary, mentioned by another poster, is much better. It uses 2i as its base, and 0-3 as its digits. It's a proper numbering system, where the length of the written number increases logarithmically with the size of the number. It can also express every complex number.
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Re: Base pi

Postby mr-mitch » Wed Aug 04, 2010 8:34 am UTC

skeptical scientist wrote:Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.


Personally I favour the reverse, the algorithm is something like:

Take value a in base X and we want it in base Y.
Divide the value a by Y in base X, and record the quotient and the remainder.
The remainder is the first digit (i.e. units). Repeat with the quotient (the next digit would be Y, then Y² and so on).

It's similar to GCD. Since the remainder is always less than Y, it works.

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Re: Base pi

Postby Mike_Bson » Sun Aug 08, 2010 7:56 pm UTC

Does anyone know what the Champernowne Constant (0.12345678910111213...) would be like in a non-natural base? In an integer base, whether that be base 10, or base 4 (where it would be 0.123101112132021...), it will always be like that (as I understand, am I wrong?). What would it do in base 1.5, or phi, or pi? Also, is there any meaning to having a negatve number base?

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Re: Base pi

Postby Talith » Sun Aug 08, 2010 8:27 pm UTC

As far as i'm aware, the constant itself is defined in base 10. If you put it into a new base, it won't have the same form as the base 10. You can of course, define a new set of constants [imath]\mathbf{Champ}=\{C(n)=0.[1]_n[2]_n[3]_n[4]_n[5]_n[6]_n..._n |n \in \mathbf{Z}\}[/imath] so that [imath]C(4)=0.123101112132021..._4=0.426111111111111028901245318..._{10}[/imath] which as you can see, doesn't have the form that you want when in base 10, but still has some of the properties of C(10) in that it is irrational and 4-normal. C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne and curiously, it hasn't been proven yet that C(n) is normal for any n other than n=10 - probably through lack of people trying. [must remember to read more thoroughly before posting]
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Re: Base pi

Postby Mike_Bson » Sun Aug 08, 2010 8:30 pm UTC

Talith wrote:As far as i'm aware, the constant itself is defined in base 10. If you put it into a new base, it won't have the same form as the base 10. You can of course, define a new constant Champernowne(n) so that [imath]C(4)=0.123101112132021..._4=0.426111111111111028901245318..._{10}[/imath] which as you can see, doesn't have the form that you want when in base 10, but still has some of the properties of C(10) in that it is irrational and 4-normal. C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne and curiously, it hasn't been proven yet that C(n) is normal for any n other than n=10 - probably through lack of people trying.

So 0.12345678910... and 0.12310111213 (base 4) are two different quantitative values? Wikipedia was a bit unclear, then. . . .

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Re: Base pi

Postby antonfire » Sun Aug 08, 2010 8:39 pm UTC

It's pretty clear that they're not all the same. Just from the first digit, C(n) is between 1/n and 2/n.
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Re: Base pi

Postby Talith » Sun Aug 08, 2010 8:41 pm UTC

Yeh, the way the constant is defined is determined entirely by what base you are working in. If you put all of the C(n) on the number line, no two would be the same and they would all be in the interval (0,1). It might be easier to see this if you work out a few in different bases and then convert them to base 10. (I'd be interested to see what a graph of C(n) looks like)

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Re: Base pi

Postby Mike_Bson » Sun Aug 08, 2010 8:58 pm UTC

Talith wrote: (I'd be interested to see what a graph of C(n) looks like)

Hm, that's a thought. I'll see what I can make up, for the fun of it.

EDIT- It's a pretty simple graph. What I noticed is that C(1) would be about 1 in base 10, C(2) would be about 0.5, C(3) would be about 0.3. I think you see the pattern; the graph for y=C(x) is about the same as the graph of y=1/x (on the right side, at least).

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Re: Base pi

Postby gmalivuk » Sun Aug 08, 2010 9:19 pm UTC

Mike_Bson wrote:EDIT- It's a pretty simple graph. What I noticed is that C(1) would be about 1 in base 10, C(2) would be about 0.5, C(3) would be about 0.3. I think you see the pattern; the graph for y=C(x) is about the same as the graph of y=1/x (on the right side, at least).
Right, because as previously mentioned it's always going to be between 1/n and 2/n.

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Re: Base pi

Postby NathanielJ » Mon Aug 09, 2010 1:42 am UTC

Talith wrote:C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne


Can you provide a source for this? Everything I've read says that it's known to be 10-normal but that the question remains open for n-normality when n \neq 10.
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Re: Base pi

Postby Talith » Mon Aug 09, 2010 1:50 am UTC

Sorry, that was my mistake. I think it was a combination of me misunderstanding notation and not remember the wiki entry correctly. I though it said C(10) was known to be normal but C(n) it wasn't know for. It turns out it says "...C10 is normal in base ten, although it is possible that it is not normal in other bases".

I'm used to 'n-normal' meaning it's normal in base n, and 'normal' meaning it's absolutely normal.


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