# Pi in Different Bases?

Discussion in 'Recreational Math' started by Pavel314, Mar 27, 2005.

1. ### Pavel314Guest

Pi has been computed in Base 10 to millions of decimal places. Does anyone
know if any work has been done in computing Pi in other bases? I Googled but
didn't find anything; any URL's you could point me to would be appreciated.
I would assume that it would have to be a non-repeating decimal in any base,
as it is an irrational number.

I was fooling around with an Excel spreadsheet and came up with the Pi
equivalencies below in Bases 2 through 16 for a limited number of digits.
For bases over 10, I use the standard notation of symbols A=10, B=11, etc.,
for digits 10 or greater.

I plugged e into my spreadsheet as a base for the final entry in the series.
Could an irrational number in an irrational base number system end up as a
repeating decimal?

BASE PI
2
11.00100100001111

3
10.01021101222201

4
3.02100333122220

5
3.03232214303343

6
3.05033005141512

7
3.06636514320361

8
3.11037552421026

9
3.12418812407442

10
3.14159265358979

11
3.16150702865A48

12
3.184809493B9186

13
3.1AC1049052A2C7

14
3.1DA75CDA813752

15
3.21CD1DC46C2B7A

16
3.243F6A8885A300

e 10.10100202000211

Paul

Pavel314, Mar 27, 2005

2. ### fiziwigGuest

<quote>
One of the most amazing mathematical results of the last few years was
the discovery of a surprisingly simple formula for computing digits of
the number pi. Unlike previously known methods, this one allows you to
calculate isolated digits-without computing and keeping track of all
the preceding numbers.

"No one had previously even conjectured that such a digit-extraction
algorithm for pi was possible," says Steven Finch of MathSoft, Inc. in
Cambridge, Mass.

The only catch is that the formula works for hexadecimal (base 16) or
binary digits but not for decimal digits. Thus, it's possible to
determine that the 40 billionth binary digit of pi is 1, followed by
00100100001110. . . . However, there's no way to convert these numbers
into decimal form without knowing all the binary digits that come
before the given string.

</quote>
from: http://www.sciencenews.org/pages/sn_arc98/2_28_98/mathland.htm

--gary shannon

fiziwig, Mar 27, 2005

3. ### Stan LiouGuest

Computation of pi in other bases has been done; in fact, one can
purchase whole CDs of binary (or, equivalently, base-2^n), which
can be used as a for pseudorandom number generation. As to how
efficient computation is done in base-2^n, Mr. Shannon's reply
goes into this.
Certainly. Pi in base pi is exactly one. However, no
transcendental number (as pi is) can be repeating in an
algebraic base. However, there is no general way to determine
whether a number is algebraic or transcedental, and just
because the base is trascedental (as e is also) does not
mean it will have repeating representation.
How about base 2i [pseudo-quaternary]? Base i-1 [pseudo-binary]?

Stan Liou, Mar 28, 2005
4. ### Walter BaeckGuest

Certainly. Pi in base pi is exactly one.

Make that ten.

--
__________
\~ALCATEL/~~~~Walter Baeck
~\~BELL~/~~~~~Alcatel Belgium
~~\~~~~/~~~~~~DSL Microelectronics Design
~~~\~~/~~~~~~~E-mail :
~~~~\/~~~~~~~~Phone : +32 3 240 73 86

Walter Baeck, Mar 28, 2005
5. ### Gottfried HelmsGuest

Am 27.03.05 21:48 schrieb Pavel314:
Only if PI and E were "algebraically dependent", so that
(let f = 1/E , and d integers>=0 )

d0 + d1*f + d2*f^2 ... di*f^(i-1)
+ f^i *( d0 + d1*f + d2*f^2 ... di*f^(i-1))
+ f^(2i)*( d0 + d1*f + d2*f^2 ... di*f^(i-1))
+ ...
= PI

[ d0 + d1*f + d2*f^2 ... di*f^(i-1) ] * ( 1+ f^i + (f^i)^2 + (f^i)^3...)
= [ d0 + d1*f + d2*f^2 ... di*f^(i-1) ] * 1/(1-f^i)

And this means that the following equation must be satisfied

PI = [ d0 + d1*f + d2*f^2 ... di*f^(i-1) ] * 1/(1-f^i)
= [ d0 + d1*f + d2*f^2 ... di*f^(i-1) ] * e^i/(e^i-1)

with finite many digits d0..di.
THis would express a rational proportion of a finite polynomial in E
to PI, which also means algebraical dependence.

Gottfried Helms

Gottfried Helms, Mar 28, 2005
6. ### Patrick HamlynGuest

Make it 10. By no means is it ten.

Patrick Hamlyn, Mar 28, 2005