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irrational-base number system

(thing)     by Ulysses       Thu Jun 15 2000 at 8:01:31

A number system that is based on an irrational number or numbers, or
is composed entirely of irrational numbers. i.e. pi, e, and the
square root of 2.

 
 
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(idea)     by Sylvar      Thu Jun 15 2000 at 10:28:35
An example would be base pi, like this:

3021[pi] = 3pi^3 + 0 + 2pi + 1

You might be asking yourself: what's the use of that? Wouldn't any
useful number have an infinite pi-cimal representation? Well, yes and
no. 0 and 1 are still 0 and 1 in any base. And if you're working with
circles, spheres, and "spheres" in more than 3 dimensions, base pi is
fairly useful. But it's true that most rational numbers would have an
infinite number of digits in base pi.

Base sqrt(2) is even more useful, since it becomes base 2 (binary) if
you set every other digit to 0. Think about it:

(a[i] x 2^i) + (a[i-1] x 2^i-1) + ... + (a[1] x 2^1) + (a[0] x 2^0)
...

is really the same as

(a[i] x sqrt(2)^2i) + (a[i-1] x sqrt(2)^2(i-1)) + ... + (a[1] x sqrt
(2)^2x1) + (a[0] x sqrt(2)^2x0) ...

and so you have a system that can comfortably be used for rational
numbers, by skipping every other digit, and can also be used to
express numbers that have more to do with fractional powers of 2.
 
 
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(thing)    by FordPrefect       Sun Jun 25 2000 at 0:18:35
The problem with base sqrt(2) is that numbers would have multiple
possible representations.

For example, I could represent decimal 12 as 1010000[sqrt(2)],
because 4+8=12. But a base conversion algorithm would give the
result: 100000000.01001000000001...[sqrt(2)] because sqrt(2)^7, or
11.313712, is closer to 12 than sqrt(2)^6, or 8. Thus, in base sqrt
(2), the number 12 is rational and irrational at the same time.

A writeup in base pi points out that this problem occurs in any non-
integral base.
 
 
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(idea)     by Seqram       Wed Sep 13 2000 at 1:38:55

Actually, FordPrefect, it isn't the fact that it's less than 2 that
causes the problem.

Consider base sqrt(5), which is greater than 2. (sqrt(5) ~ 2.236).
Take decimal 280, which is 2010100[sqrt(5)]. But the usual conversion
routines will give 10000000.1001...[sqrt(5)], since sqrt(5)^7 is
279.51, which is closer to 280 than 2*sqrt(5)^6, or 250.

Same for base sqrt(10), which in some ways is easier to work in, and
is greater than 3. 322 decimal is "obviously" 30202[sqrt(10)], but
it's also 100012.12...[sqrt(10)], by similar reasonings.

I'm thinking it may also have to do with the way the base^0 place
works. Ponder: the equations 10-1=(b-1) (where b-1 is the digit for
one less than the base b) works for all integer bases, but does not
work for irrational bases. The problem is that we're using integer
multiples of integer powers of the base, and when the base isn't an
integer that isn't so well-behaved anymore.

 
 
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"42" is      Feature request                             Fibonacci
even to any  for Universe 2.0   base pi                  base
base

Deathmatch:  interesting        How to determine
e vs pi      number bases       whether a number is      e
                                even or odd in any base

             Also starring a                             irrational
Bluetooth    telephone as "The  All bases are base 10    number
             Telephone"

Base nine                                                Dozenal
number       base 20            False!                   Society of
system                                                   America

pi           base e             Memory Base 128          well-behaved

Killer       The Base project   base 2 prefixes          integral
Instinct

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