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Compute pi to base 12 using Python?

 
 
Dick Moores
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      04-13-2005
I need to figure out how to compute pi to base 12, to as many digits as
possible. I found this reference,
<http://mathworld.wolfram.com/Base.html>, but I really don't understand
it well enough. Could someone show me how to do what I need?

Thanks,

Dick Moores
http://www.velocityreviews.com/forums/(E-Mail Removed)

 
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Bengt Richter
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      04-13-2005
On Wed, 13 Apr 2005 02:06:11 -0700, Dick Moores <(E-Mail Removed)> wrote:

>I need to figure out how to compute pi to base 12, to as many digits as
>possible. I found this reference,
><http://mathworld.wolfram.com/Base.html>, but I really don't understand
>it well enough. Could someone show me how to do what I need?
>
>Thanks,
>
>Dick Moores
>(E-Mail Removed)
>

See if this is enough digits for homework?

(Digits A and B are 10 and 11 in decimal respectively)

3184809493B918664573A6211BB151551A05729290A7809A49 2742140A60A55256A0661A03753A3AA54805646880181A
3683083272BBBA0A370B12265529A828903B4B256B8403759A 71626B8A54687621849B849A8225616B442796A31737B2
29B2391489853943B8763725616447236B027A421AA17A38B5 2A18A838B01514A51144A23315A3009A8906B61B8B48A6
2253A88A50A43BA0944572315933664476B3AABB7758397512 0683526B75B462060BB03B432551913772729A21475535
31793848A0402B999B5058535374465A68806716644039539A 8431935198527B9399B112990ABB0383B107645424577A
51601B3624A88B7A676A3992912121A213887B92873946A613 32242217AA7354115357744939112602BA4B888818A326
9222B528487747839994AB223B65B8762695422822669BA00A 586097842A51750362073B5A768363B21BB1A97A4A1944
47749399804922175A068A46739461990A2065BB0A30BBAB70 24A585B1A84428195489784A07A331A7B0A1574565B373
B05B03A5A80A13AB87857734679985558A5373178A7B282719 92A3894A5776085083B9B238B2220542462888641A2BAB
8B3083AB49659172A312B78518654494A068662586A181835A 64440B2970A12281397589881536720890580103288144
9223841428763329617531239B9A657405584014534390B587 625606BB80923795944B43757A431B039556282978A6A4
9590553490BA1844947175637A908247B50127722464441380 A852B0847B5813019BB70A67663B426565434069884476
132193344BA55A2128A03838974606B851B2979321A4080672 25A5AA4B3464A1A17473595333909AB9127079655B3164
B68B9B28A9B818A220A025AB0934203995B7A62A7AA7393553 40539BA3182905B193905603A43B660B9426A922946971
44A896A5B2339358BB2B7294BB89635B071A6351211360B820 B1882AB8433B54757B87A373284B1BA182A10326476B36
9A4A6365B58B8018994BB152556765475A704BB94B6B2A3945 8971A8B90512786B5029404818644323552916170B3ABB
7363496427B088B68725A68570040617949289077B278069A0 9B559324B8A66828B40549B0296065B2300330592569A7
B76B92BA1293585B6A9B604567A0901362856373B4B5689794 6256B4172B1B50474351364749A33996A81BA8847347A8
411B850B79A03018291672AA0945656A159AA6AA0A845531A5 92005B8A34366B882257107B190969A846474836A98007
50778920BA797297A2791101B0685A86BB704B9BAA17B05529 3679843B35215B0A8B1182B611953B080AA5431B219907
A8448A81B1A9493245676B88013B4703352408595941586210 14216619553246570601967448B470174B924489244481
7453865A4003B5AA7176451AAB90681A949786154AA0404773 82BA69371041710B8728458A23979252B254236753A44A
1900AA283536A227648812525743868B410A567794663359A6 726A5286783328135114789B7645505B047848020A730A
9557B206776AA56A19682744107901306B29008808619866B4 911A05264B872A46B5376383932699531B449195640B62
A63622830886247A47B3957169861239358041AA281333622A A15912B0A636047A489BB0726282A78B96671B27305A96
52496B9B999011A7BA36898891665B1A6009058978850A21B0 1A158A1473B84A192B8672542A2A7056581995207A436A
5B3BA2824637A3112ABB57176468206A071200A327B3216425 148100786502AA21236ABB35499277670A126973058340
3B1922A483856007301983989159BB688A58B602339806B630 02A339A50B0BA533B84827793913081070A32595A10180
3A9A20234691B1A0B623274B69B0B44688195169461059543A 252BB05208720BA13118266A872B26B9B584959B451795
19534B221A335A2BB6971B3276B3A63A5B791723109B176529 BB90651584279B7825712521B8269800738B07A62B1454
7884414451224092937165625696557A78799A82126613535A 36B410309B759976119777B895801074B9779B9B513753
8B2799951012273399BB818B721967957713B90947B2A11A6A 665848B22B531726616515939323229080B8AB574AA749
4773AB411A57150203067A112944833235A86153803A98689A 0762B79835413A46B347888A4AAB259665694B93129B62
1391751430A98B002620718437A7B85B891179479651AA3410 663715415B55BA47AA59465AB81567B7655780A8038135
85230122578485B071A529692B19A6537B28616A6355681694 5380634A90470354AAB303884B7B09B2037B95405BA145
704B19B14AA8028810881AB6072441194A875477836B37704B 5199062319A336375437403562A663B835B891957883AB

Hint: Lambert Meertens. Tweak the algorithm you find

Regards,
Bengt Richter
 
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Dick Moores
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      04-13-2005
Bengt Richter wrote at 03:19 4/13/2005:
>On Wed, 13 Apr 2005 02:06:11 -0700, Dick Moores <(E-Mail Removed)> wrote:
>
> >I need to figure out how to compute pi to base 12, to as many digits as
> >possible. I found this reference,
> ><http://mathworld.wolfram.com/Base.html>, but I really don't understand
> >it well enough. Could someone show me how to do what I need?
> >
> >Thanks,
> >
> >Dick Moores
> >(E-Mail Removed)
> >

>See if this is enough digits for homework?


This is not homework, nor am I a student, though I am trying to learn
Python. I'm just trying to help an artist acquaintance who needs (I just
learned) the first 3003 digits of pi to the base 12.

>Hint: Lambert Meertens. Tweak the algorithm you find


Sorry. Your hint is beyond me.

Dick Moores
(E-Mail Removed)


 
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Roman Neuhauser
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      04-13-2005
# (E-Mail Removed) / 2005-04-13 03:27:06 -0700:
> Bengt Richter wrote at 03:19 4/13/2005:
> This is not homework, nor am I a student, though I am trying to learn
> Python. I'm just trying to help an artist acquaintance who needs (I just
> learned) the first 3003 digits of pi to the base 12.
>
> >Hint: Lambert Meertens. Tweak the algorithm you find

>
> Sorry. Your hint is beyond me.


it says "use google".

--
How many Vietnam vets does it take to screw in a light bulb?
You don't know, man. You don't KNOW.
Cause you weren't THERE. http://bash.org/?255991
 
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Dan Bishop
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      04-13-2005
Dick Moores wrote:
> I need to figure out how to compute pi to base 12, to as many digits

as
> possible. I found this reference,

<http://mathworld.wolfram.com/Base.html>,
> but I really don't understand it well enough.


How many stars are in "*************************"?

You probably answered "25". This means that, for convenience, you've
broken down the row of stars into ********** + ********** + *****, that
is, 2 tens with 5 left over, which the base-10 numeral system denotes
as "25".

But there's no reason other than tradition why you should arrange them
into groups of 10. For example, you could write it as ******** +
******** + ******** + *, or 3 eights plus 1. In octal (base-
notation, this is written as "31"; the "tens" place in octal represents
eights.

In general, in the base-r numeral system, the nth digit to the left of
the ones digit represents r^n. For example, in the binary number
11001, the place values for each digit are, right to left, 1, 2, 4, 8,
and 16, so the number as a whole represents
116+18+04+02+11=16+8+1=25. This analogous to 25=210+5 in
base-10.

It's also possible to write it as 38+04+02+11 = 3001 base 2,
but by convention, base-r only uses the digits in range(r). This
ensures a unique represenation for each number. This makes "11001" the
unique binary representation for decimal 25.

Note that for bases larger than 10, the digits will be numbers that are
not single digits in base 10. By convention, letters are used for
larger digits: A=10, B=11, C=12, ... Z=35. For example, the number
(dec) 2005 = 112+112+1112+11 is represented in base-12 by
"11B1".

Fractions are handled in a similar manner. The nth place to the right
of the radix point (i.e., the "decimal point", but that term is
inaccurate for bases other than ten) represents the value radix**(-n).

For example, in binary,
0.1 = 1/2 = dec. 0.5
0.01 = 1/4 = dec. 0.25
0.11 = 1/2 + 1/4 = 3/4 = dec. 0.75
0.001 = 1/8 = dec. 0.125
0.01010101... = 1/4 + 1/16 + 1/64 + ... = 1/3
0.0001100110011... = 1/10 = dec. 0.1

The last row explains why Python gives:

>>> 0.1

0.10000000000000001

Most computers store floating-point numbers in binary, which doesn't
have a finite representation for one-tenth. The above result is
rounded to 53 signficant bits
(1.10011001100110011001100110011001100110011001101 02^-4), which is
exactly equivalent to decimal
0.100000000000000005551115123125782702118158340454 1015625, but gets
rounded to 17 significant digits for display.

Similarly, in base-12:

0.1 = 1/12
0.14 = 1/12 + 4/144 = 1/9
0.16 = 1/12 + 6/144 = 1/8
0.2 = 2/12 = 1/6
0.3 = 3/12 = 1/4
0.4 = 4/12 = 1/3
0.6 = 6/12 = 1/2
0.8 = 8/12 = 2/3
0.9 = 9/12 = 3/4
0.A = 10/12 = 5/6

Notice that several common fractions have simpler representations in
base-12 than in base-10. For this reason, there are people who believe
that base-12 is superior to base-10.
(http://www.dozenalsociety.org.uk)

> Could someone show me how to do what I need?


You'll need 3 things:

(1) An algorithm for computing approximations of pi.

The simplest one is 4*(1-1/3+1/5-1/7+1/9-1/11+...), which is based on
the Taylor series expansion of 4*arctan(1).

There are other, faster ways. Search Google for them.

(2) An unlimited-precision numeric representation. The standard
"float" isn't good enough: It has only 53 bits of precision, which is
only enough for 14 base-12 digits.

The "decimal" module will probably work, although of course its base-10
internal representation will introduce slight inaccuracies.

(3) A function for converting numbers to their base-12 representation.

For integers, this can be done with:

DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
def itoa(num, radix=10):
is_negative = False
if num < 0:
is_negative = True
num = -num
digits = []
while num >= radix:
num, last_digit = divmod(num, radix)
digits.append(DIGITS[last_digit])
digits.append(DIGITS[num])
if is_negative:
digits.append("-")
digits.reverse()
return ''.join(digits)

For a floating-point number x, the representation with d "decimal"
places count be found by taking the representation of int(round(x *
radix ** d)) and inserting a "." d places from the right.

 
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Dick Moores
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      04-13-2005
Dan Bishop wrote at 04:07 4/13/2005:
>(3) A function for converting numbers to their base-12 representation.
>
>For integers, this can be done with:
>
>DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
>def itoa(num, radix=10):
> is_negative = False
> if num < 0:
> is_negative = True
> num = -num
> digits = []
> while num >= radix:
> num, last_digit = divmod(num, radix)
> digits.append(DIGITS[last_digit])
> digits.append(DIGITS[num])
> if is_negative:
> digits.append("-")
> digits.reverse()
> return ''.join(digits)


I see this works perfectly for integers. Thanks!

>For a floating-point number x, the representation with d "decimal"
>places count be found by taking the representation of int(round(x *
>radix ** d)) and inserting a "." d places from the right.


But I'm sorry, but I can't follow you. I do have the first 10000 or so
places of pi base 10 (<http://mathwithmrherte.com/pi_digits.htm>), but
could you show me what to do with, say, just 3.14159?

I apologize for being so dense.

Dick Moores
(E-Mail Removed)


 
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Roy Smith
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      04-13-2005
In article <(E-Mail Removed). com>,
"Dan Bishop" <(E-Mail Removed)> wrote:

> But there's no reason other than tradition why you should arrange them
> into groups of 10.


Well, it is traditional for people to have 10 fingers

Other fun things to think about are negative bases. For example, 3(10) =
111(-2). That's 1*(-2)^2 + 1*(-2)^1 + 1*(-2)^0 = 4 - 2 + 1. I can't
think of any use for negative bases, but they are a fun game to play with
(if you're into that sort of stuff).

Non-integer bases are fun too.
 
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Dan Bishop
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      04-13-2005
Dick Moores wrote:
> Dan Bishop wrote at 04:07 4/13/2005:

....
> >For a floating-point number x, the representation with d "decimal"
> >places count be found by taking the representation of int(round(x *
> >radix ** d)) and inserting a "." d places from the right.

>
> But I'm sorry, but I can't follow you. I do have the first 10000 or

so
> places of pi base 10 (<http://mathwithmrherte.com/pi_digits.htm>),

but
> could you show me what to do with, say, just 3.14159?


First, decide how many "decimal" places to use for the conversion.
Five decimal digits is equivalent to 5*log(10)/log(12) = 4.63 base-12
digits, so use 4 digits.

Next, multiply by 12**4, obtaining the value 65144.01024, and round to
the nearest integer, 65144. Convert this to base 12, obtaining 31848.
But this is 12**4 times the number we really want, so divide this by
12**4 (i.e., shift the radix point left 4 places), for a final result
of 3.1848.

 
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Scott David Daniels
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      04-13-2005
Roy Smith wrote:
> In article <(E-Mail Removed). com>,
> "Dan Bishop" <(E-Mail Removed)> wrote:
>
>
>>But there's no reason other than tradition why you should arrange them
>>into groups of 10.

>
>
> Well, it is traditional for people to have 10 fingers
>
> Other fun things to think about are negative bases. For example, 3(10) =
> 111(-2). That's 1*(-2)^2 + 1*(-2)^1 + 1*(-2)^0 = 4 - 2 + 1. I can't
> think of any use for negative bases, but they are a fun game to play with
> (if you're into that sort of stuff).
>
> Non-integer bases are fun too.


If you think those are fun, try base (1j - 1)

--Scott David Daniels
(E-Mail Removed)
 
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Roy Smith
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      04-13-2005
Scott David Daniels <(E-Mail Removed)> wrote:
>If you think those are fun, try base (1j - 1)


Get real. I can't imagine using anything so complex.
 
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