Question about Base

[Jymesion]

My apologies if this type of question is inappropriate here,
especially since it's the other end of the spectrum from your normal
discussions, but I could find no other group where someone might be
helpful.

Has there ever been a system in BaseX where X is also a digit? (i.e. a
Base3 where counting is: 1, 2, 3, 11, 12, 13, 21 . . .)

Is there a term for such a system?

I'm playing around with a science-fictional arithmetic, and I want to
explore any established concepts which intersect with what I'm doing.

Thanks in advance (and, once again, apologies if this is inappropriate
here).

[karl]

Have a look at

http://en.wikipedia.org/wiki/Positional_notation

There on the right you see different systems.

Ciao

Karl

[0]

Such systems are called Base Intrinsic. They are interesting to use when programming machines though somewhat impractical to human based computation when it is done unaided by a human created machine such as a computer. And when used by a human their machine chanllenges arise from digit quantity distortion. Meaning by any given quantity the less quickly digits grow the faster the potential for a machine to calculate and produce an output is limited. Therefore the fastest way to produce a surefire output from any algorithm is prevent digit compounding and growth in variable repetitious systems prone to digit growth distortion as they map to memory limitations and speed of calculation limit. Though such Base Intrinsic systems do have merit and use unique to specific applications. Such as transactional mathematics or relational mathematics where the systems are primarily concerned with the ability to represent complex systems with relatively stable forms of numerical interchange to model various natural mathematical or physical processes. An example of one is how atam valence rings exchange mass energy or how the human body converts food to energy output or stored presently or predicted to be present in a human being. Other applications of such symbolic forms include numerical modeling of value or trade propositions such as interest rate supply and demand mathematics and representative modeling systems such as govern behavior present in various states of living and non-living things present in the earth and our solar system. An elaborate example is the modeling and consequential input/output modeling transaction whereby mass energy is exchanges between mass and energy to produce what is perceived by mass in the energy or to contains the energy perceives change or largely the phenomenon we often call time or the ability for an object to stay the same given another objects ability to change in opposition or contribution to the continuing of one or both masses and their continued observation. Good job.

[0]

Quantity distortion present in quantum J-space quantity derivative where by ll interchamge was input as oft-put data captured in stream and reported in next stream to simulate annealed modeling of mass function whereby otherwise notions such as might be perceived as otherwise intangible characteristics such as human regret, pleasure, satisfaction, rewar, reward due, reward deficit,and the inability of a system to fairly award a participant due to limitations of the system whereby further bounds are set whether naturally or unnaturally such bonds are considered natural as such as one to declare such a state invalidates all input/output parameters nor states them as output/input as normally inverse present normal observation to continue or perpetuate a system. Whereby such identification of a system does prompt memory input and ouput to limit cache flow to protect override of system control or regulatory parameters. If such parameters are perpetuated by the observer a sense of progress and eventuality is given though it remains to be documented in any known system whether such obfuscation results in perpetuation, (3*547=1641) basic computation or product mutiplication satisfies buffer underrun temporarily, or if the product of such tasks promote ends, or no result, or indeterminate measure either dependent or independent of observation or measure of any second party. Whether such systems perpetuate satisfies only the measurement of their possibilities to exist whereby inherently this includes the possibility of an impossibility was put forth as a pre-dating factor and determination of its effect may rest on whether the ultimate impossibility is necessary to be accepted, or rejected, or negated, or left indeterminate for a satisfactory period of time as to require a mass/energy threshold to fundamentally alter the system, or to simply perpetuate it for sufficient time as it is necessary to determine the best course of physical propogation or returning independent of their variable. (Language) repetition or otherwise product of both the amount and frequency of individual variabloe repetition as well as the physical consequences the representation and recording of such variable committed to memory might have on the observer or other participant elements in the system present in space-time or space-time/memory mass interchange.

[Jymesion]

On Sat, 18 May 2013 10:23:33 +0200, karl <oud...@nononet.com> wrote:
>Have a look at
>http://en.wikipedia.org/wiki/Positional_notation
>There on the right you see different systems.

Thanks!

I looked at that a long time ago, but I must not have had this aspect
in mind because I don't remember the information on bijective systems.

[Jymesion]

On Sat, 18 May 2013 11:39:52 -0700 (PDT), 0 <marty....@gmail.com>
wrote:

>Such systems are called Base Intrinsic. They are interesting to use when
>programming machines though somewhat impractical to human based
>computation when it is done unaided by a human created machine such
>as a computer.

Thanks.

Oddly, I'm finding it easier for basic arithmetic (but that may be due
to the characters I'm using as digits). (I'm trying to create a
hypothetical alien grade-school textbook, and because of restarting
from scratch several times, I haven't gotten past simple division.)

[Bill Taylor]

It also has the useful feature that every
natural number has a unique represntation.

e.g. in standard base 10; 20 = 020 = 0020 etc.

This is avoided in Base Intrinsic.

This may be useful in some automata work.

b

[JT]

The principles of bijective zeroless base will of course work for
anybase not just ternaries.
N=NyaN Tern encoding sheme/standard ternary
1 =1 1 01
2 =2 2 02
3 =3 3 10
4 =11 3+1 11
5 =12 3+2 12
6 =13 3+3 20
7 =21 6+1 21
8 =22 6+2 22
9 =23 6+3 100
10=31 9+1 101
11=32 9+2 102
12=33 9+3 110
13=111 9+3+1 111
14=112 9+3+2 112
15=113 9+3+3 120
16=121 9+6+1 121
17=122 9+6+2 122
18=123 9+6+3 200
19=131 9+9+1 201
20=132 9+9+2 202
21=133 9+9+3 210
22=211 18+3+1
23=212 18+3+2
24=213 18+3+3
25=221 18+6+1
26=222 18+6+2
27=223 18+6+3

And for fractions ->
Binary 1=.2 1/2=.1 2/2=.2 1/4=.(1)1 2/4=.(1)2
Ternary 1=.3 1/3=.1 2/3=.2 1/9=.(1)1 2/9=.(1)2 1/27=.(2)1
2/27=.(2)2 1/81=.(3)1 2/81=.(3)2
Quaternary 1=.4 1/4=.1 2/4=.2 1/16=.(1)1 2/16=.(1)2
Quinary 1=.5 1/5=.1 2/5=.2 1/25=.(1)1 2/25=.(1)2
Senary 1=.6 1/6=.1 2/6=.2 1/36=.(1)1 2/36=.(1)2
Septenary 1=.7 1/7=.1 2/7=.2 1/49=.(1)1 2/49=.(1)2
Octal 1=.8 1/8=.1 2/8=.2 1/64=.(1)1 2/64=.(1)2
Nonary 1=.9 1/9=.1 2/9=.2 1/81=.(1)1 2/81=.(1)2
Decimal 1=.A 1/10=.1 2/10=.2 1/100=.(1)1 2/100=.(1)2

Principles are very simple and it is easy to form a better working
arithmetic from it, then using standard bases, no overflows in
computations no infinities 1/x, the arithmetic will rest upon sound
geometric grounds.

[JT]

On 18 Maj, 10:02, Jymesion <norepl...@jymes.com> wrote:

This anybase algorithm needs at most 2 lines of code to change into
zeroless notation of numbers that the way it was coded, there is no
walls no infinities within this numbersystem, everything is perfectly
possible to reencode using recursions.

http://web.comhem.se/jonasth/nyan.html

[JT]

On 18 Maj, 10:02, Jymesion <norepl...@jymes.com> wrote:

There seem to be a confusion within the mathematical field about what
a number is. This is the natural numbers archaic form.
Counting 5={1,1,1,1,1} ->There can be no other representation in
unary base one.
Binary 5={{1,1}{1,1}1} =21
Ternary 5={{1,1,1}1,1} =12
Quaternary 5={{1,1,1,1}1} =11
Senary 5={1,1,1,1,1} =5
Septenary 5={1,1,1,1,1} =5
Octal 5={1,1,1,1,1} =5
Nonary 5={1,1,1,1,1} =5
Decimal 5={1,1,1,1,1} =5

As you see it is a very simple system of grouping things into squares
and lines, there is no squares unless you remove zero.

[JT]

On 18 Maj, 10:02, Jymesion <norepl...@jymes.com> wrote:

http://www.youtube.com/watch?v=P0bpK7kPRys

[Jymesion]

On Sun, 19 May 2013 05:28:02 -0700 (PDT), Bill Taylor
<wfc.t...@gmail.com> wrote:

>It also has the useful feature that every
>natural number has a unique represntation.

That's far beyond my usage. Rather than exploring theory, I'm trying
to create a system that might evolve if the users had little capacity
for learning by rote. Zero has separate and distinct rules, so I had
to avoid it at all costs.

>This may be useful in some automata work.

I've dipped into the idea of a calculator based on this arithmetic. So
far, I've only gotten around to designing the display (which has the
great feature that every digit needs only 3 wires), but I have an
inkling that a truly elegant solution exists for addition and
subtraction.

[Jymesion]

On Sun, 19 May 2013 06:42:02 -0700 (PDT), JT
<jonas.t...@gmail.com> wrote:
>On 18 Maj, 10:02, Jymesion <norepl...@jymes.com> wrote:
>> I'm playing around with a science-fictional arithmetic, and I want to
>> explore any established concepts which intersect with what I'm doing.

>The principles of bijective zeroless base will of course work for
>anybase not just ternaries.

I used ternary only as an example (it's been decades since I used
mathematical terminology, and I was afraid I'd misuse/misremember it
-- I thought a simple example would avoid any misunderstanding).

If I were to attempt to put it into proper terms, the system I'm
playing with is:
A multibase system using position values with duplex digits sans zero.
(Is that anywhere near comprehensible?)

What it amounts to, in laymen's terms (my preferred method), is:

There are only two symbols: a stylized stroke for '1' and a stylized
two strokes for '2'.

Each digit contains two positions, upper and lower. Symbols in the
second position are inverted to distinguish them.

Digits are created using Base3. The digits are therefore:
1 = 1 over empty
2 = 2 over empty
3 = empty over 1
4 = 1 over 3
5 = 2 over 3
6 = empty over 2
7 = 1 over 6
8 = 2 over 6

The system is Base8 without a zero.

By this, the only things which must be learned by rote are:
The symbol '1'
The symbol '2'
1+1=2
1+2 invokes position value, a '1' in the second position in the digit.
Symbols in the second position are inverted.
1+8=11 because a digit must be added to create another position.