1/3 unfinished decimal expansion.

JT

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Feb 1, 2013, 12:32:46 AM2/1/13

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How is fraction 1/3 expressed in base 3, of course in base 10 it do
have unfinished decimalexpansion.
But has it really in base 3???
And this is where number zero gets interesting, because when you use
zero number 3 is expressed 10 in base 3 right?
But in a number system without 0 it should *undoubtly* be expressed a
single 3.
And number 6 be 13 number 9 be 23 do you people agree?
From this (may) also follow that 1/3 in such a number system would be
expressed .1 do you people agree?
And 0.999... would be expressed as 0.3 or.......?
I could easily adjust my base changer to this, and in this zeroless
number system 3+1/3 would be expressed 3,1
6+1/3 would be expressed 2,1
Now can you people see any benefits from my new bases without 0. It
seem that unfinished decimal expansion vanish in certain bases or?
How does this basesystem make you feel angry, annoying or just
uninteresting?
Is there something deep profound to all this, if i just could
remember
what. I think it has something todo with factoring.
For now i just wants commenting upon the proposed new bases without
zeros good or bad, advantages disadvantages?
Would you like me make a version of my base changer where 1/3 do not
have unfinished decimal expansion and all bases expressed without
zeroes?

Message has been deleted

JT

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Feb 1, 2013, 12:47:41 AM2/1/13

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Below is a standard base changer that change to anybase, with two or
three lines code we could finish the tyranny of unfinished decimal
expansion forever what do you say people shall we give it a go?
http://www.youtube.com/watch?v=qItugh-fFgg
<HTML><HEAD><TITLE>TEST</TITLE>
<META http-equiv=Content-Type content="text/html;
charset=windows-1252">
<SCRIPT language=Javascript>
function anybase() {
fullnumber=document.doit.dnumb.value;
twoparts = fullnumber.split(",");
bas=document.doit.bval.value;
for(j=0;j<2;j++){
btwo=1;
bnr=0;
unr=0;
decimalnumber=twoparts[j];
while (btwo<decimalnumber){
btwo=btwo*bas;
bnr++
}
basestring="";
while (decimalnumber>0){
unr++;
set=0;
for(i=bas;i>0;i--){
st=btwo*i;
if (decimalnumber>=st) {
decimalnumber=decimalnumber-st;
set=1;
basestring=basestring+i+",";
}
}
if(set==0 && decimalnumber!=0)basestring=basestring+0+",";
btwo=btwo/bas;
}
while (bnr>=unr){
basestring=basestring+0+",";
unr++;
}
if (j==0) {radixout=basestring;document.doit.radix.value=radixout;}
if (j==1) {radixout=basestring;document.doit.fracted.value=radixout;}
}

control=document.doit.radix.value;
controldec=document.doit.fracted.value;
show=document.doit.dnumb.value+"=";
//Control Radix by go back to decimal
darr = control.split(",");
decarr = controldec.split(",");
darr.reverse();
decarr.reverse();
total=0;
totalfrac=0;
multiple=1
for (i=1;i<darr.length;i++){
cval=darr[i];
cval=cval*multiple;
total=total+cval;
show=show+"("+multiple+"*"+darr[i]+")+";
multiple=bas*multiple;
}
multiple=1
for (i=1;i<decarr.length;i++){
cval=decarr[i];
cval=cval*multiple;
totalfrac=totalfrac+cval;
show=show+"("+multiple+"*"+decarr[i]+")+";
multiple=bas*multiple;
}
totalfull=total+","+totalfrac;
document.doit.controlout.value=totalfull;
document.doit.overview.value=show;
}

</SCRIPT>
<HTML><BODY>
<FORM NAME=doit onSubmit="anybase(); return false;">
CHANGE BASE:
<input type=submit value="GO!"><P>
DECIMAL VALUE:<TEXTAREA NAME=dnumb COLS=30
ROWS=1>777777777777,3333333</TEXTAREA><BR>
BASE:<TEXTAREA NAME=bval COLS=30
ROWS=1>333</TEXTAREA><BR>
RADIX OUT:<TEXTAREA NAME=radix COLS=200
ROWS=1></TEXTAREA><BR>
FRACTION OUT:<TEXTAREA NAME=fracted COLS=200
ROWS=1></TEXTAREA><BR>
BACK TO DECIMAL CONTROL CONVERSION->
<TEXTAREA NAME=controlout COLS=30
ROWS=1></TEXTAREA><BR>
Radix HOWTO EXPLAIN<TEXTAREA NAME=overview COLS=200
ROWS=1></TEXTAREA><BR>
</FORM>
</BODY>
</HTML>

Jeffrey Goldberg

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Apr 8, 2013, 3:45:04 PM4/8/13

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On 2013-02-01, JT <jonas.t...@gmail.com> wrote:

> How is fraction 1/3 expressed in base 3,

0.1

> And this is where number zero gets interesting, because when you use
> zero number 3 is expressed 10 in base 3 right?

Right.

> But in a number system without 0 it should *undoubtly* be expressed a
> single 3.

Well, I need to know more about the system,

> And number 6 be 13 number 9 be 23 do you people agree?

You seem to be using a place value system but without 0. You
need to spell out the details. Though I can imagine it. you count

1 2 3 11 12 13 21 22 23 31 32 33 111

> From this (may) also follow that 1/3 in such a number system would be
> expressed .1 do you people agree?

OK, I'll buy that for now.

> And 0.999... would be expressed as 0.3 or.......?

Well you couldn't say "0.3" becasue you don't have "0" in your
system. So maybe you would just use ".3".

> Now can you people see any benefits from my new bases without 0.

Nope.

> It
> seem that unfinished decimal expansion vanish in certain bases or?

We already know that different bases have different properties for
terminating or non-terminating expansions of rational numbers. It
has to do with (powers of) the factors of the base.

Consider base 10. We get terminating expansions for n/2, n/4, n/8,
... and n/5, n/25, n/125. We also get terminationg expansions for
n/2*5, n/4*5, n/8*5, ... and n/2*5, 2*25, and of course n/4*25,...

So we get terminating expansions for all and only rationals that are
n/(2^i * 5^j) where i and j >= 0. 2 and 5 here are not a
coincidence. They are the prime factors of the base, 10.

So if you want a base that will give you more terminating
expansions, use base 30 (2*3*5). But 1/7 in base 30 will not
terminate. So you could use base 210 (that's a lot of symbols), and
that will get you nice terminating expensions for every denominator
that doesn't have a prime factor greater than 7. (So 1/11 will be
your first problem).

There will always be some base for which any particular expansion of
a rational number is terminating, but there is no base in which
every rational number will be terminating.

I suspect that the Babylonians use of base 60 was in part because
they could do math with ratios more easily with a base that is
2*2*3*5, but I'll leave that to historians who can probably do more
than just speculate as I have.

> How does this basesystem make you feel angry, annoying or just
> uninteresting?

The notion that whether a expansion of a rational number terminates
or not depends on the base is a good intution. Getting rid of 0,
though, just complicates things for no good reason.

> Is there something deep profound to all this, if i just could
> remember
> what. I think it has something todo with factoring.

I've spelled that out above. I'm not sure whether it counts as deep
and profound.

> For now i just wants commenting upon the proposed new bases without
> zeros good or bad, advantages disadvantages?

Getting rid of the zeros complicates the definition of the place
system and will make arithmatic must more complicated (try doing a
few multi-digit multiplications or divisions in your system). And it
gives you nothing of value.

Cheers,

-j

--
Jeffrey Goldberg http://www.goldmark.org/jeff/
I rarely read top-posted, over-quoting or HTML postings.
Reply-To address is valid.

Piergiorgio Sartor

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Apr 8, 2013, 4:08:11 PM4/8/13

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On 2013-04-08 21:45, Jeffrey Goldberg wrote:
[...]

>> For now i just wants commenting upon the proposed new bases without
>> zeros good or bad, advantages disadvantages?
>
> Getting rid of the zeros complicates the definition of the place
> system and will make arithmatic must more complicated (try doing a
> few multi-digit multiplications or divisions in your system). And it
> gives you nothing of value.

In addition to this, I would like to add a couple
of comments.

First, zero is considered one of the greatest
inventions of all the time, similar to the wheel.

Second, I suspected the equation (in "x") A+x=A
does not have a solution, in any base, without
the definition of zero.

Of course, one can call "gremlin" the value of
"x" for which that equation has a solution, but I
guess this will have to be included into the base
system and it will look like exactly as zero.

bye,

--

piergiorgio

Noob

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Apr 9, 2013, 7:21:31 AM4/9/13

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Piergiorgio Sartor wrote:

> First, zero is considered one of the greatest
> inventions of all the time, similar to the wheel.

It CANNOT be a coincidence that "zero" and "the wheel"
share the same shape!! There must be some dark forces
at work!!

Piergiorgio Sartor

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Apr 9, 2013, 12:38:45 PM4/9/13

to

On 2013-04-09 13:21, Noob wrote:
[...]

> It CANNOT be a coincidence that "zero" and "the wheel"
> share the same shape!! There must be some dark forces
> at work!!

For sure, the same shape (in 2D) as the Death Star!!!

bye,

--

piergiorgio

Andrew Swallow

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Apr 9, 2013, 2:47:50 PM4/9/13

to

Do not be surprised if you investigations come to naught. ;)

Andrew Swallow

Jeffrey Goldberg

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Apr 13, 2013, 12:10:29 AM4/13/13

to

Nothing to see here.

JT

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Apr 13, 2013, 7:41:31 PM4/13/13

to

On 13 Apr, 06:10, Jeffrey Goldberg <nob...@goldmark.org> wrote:

True zero is the invisible pink elephant of your imagination it can
not be seen.

WT Shaw

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Apr 14, 2013, 6:59:01 AM4/14/13

to

On Thursday, January 31, 2013 11:32:46 PM UTC-6, JT wrote:
.....

>
> For now i just wants commenting upon the proposed new bases without
>
> zeros good or bad, advantages disadvantages?
>
> Would you like me make a version of my base changer where 1/3 do not
>
> have unfinished decimal expansion and all bases expressed without
>
> zeroes?

When you try to force a new base to conform to old syntax, you escape some of the virtues available to any new base whereas the set of characters in any specific permutation is the simple statement defining the base. Consider that the decimal point is named specifically for the base to which is to apply and as it might be used as a pattern for some other as it need not do so. Some places use a comma instead of a period for the same purposes we use a decimal point, even use different names for higher values of numbers. There is no standard set of characters for any base, merely historic conventions as a product of a time when they were considered useful and by those that advanced some narrow use for those same conventions. It's all a matter of chosen coding and nothing more..

JT

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Apr 15, 2013, 5:58:23 AM4/15/13

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The thought up numberline is made of hashmarks, and you have to
partition it to make any sense.
Since the reals is not baseless, but fractions are.

If you wanted to represent a binary 5 as 101 ={{{1111}}1}
that just i hope you can interpretat it, here you have the collection
of hashmarks partitioned into binary base.
There is loss in decomposition of the collection, this will ill
grove weirder with the number of zeros because lack of decomposition
of the squares, you are left with a freaking huge collection that
may
or may not be a square.

Here is another way to partition the hashmarks into bases.

So as you may have guessed
Base 1 5={1,1,1,1,1} unary just hashmarks 5 using symbol
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

Also notice how the parenthesis denote the digitplace, you see i have
no problem to show how the unary full collection is grouped into
subsets making the base for our symolic representation afterall
1,2,3,4,5,6,7,8,9... and so on are just groupings that our brain
learned howto work with in reality my hashmarks are there. And they
are the only archaic option to symbols.

1,2,3,4,5,6,7,8,9
Have no other archaic representation then
1,11,111,1111,11111,111111,1111111, 11111111,111111111
Unless you chose to partition the collection into subsets
representing
digitplaces which you can see i have done above in thread.

I have not thought about representing base1(unary?) fractionals
becase fractionals is superior to partitioning into base, but
possible .{1}
for 1/3 and 3/9 .{{1,1,1} 9/27 .{{{1,1,1,1,1,1,1,1,1} what do you
think. This is the true nature of numbers collections and cuts, the
number line is just dreamed up. Numbers are baseless we partition
and
create the semantics the collections is interpretated in, not the
other way around. And i show you the simples semantics for numbers.
Collections and cuts.

So when you decode the hashmarks into a zeroless base you will get the
new Naturals
1 =1
2 =2
3 =3
4 =11 3+1
5 =12 3+2
6 =13 3+3
7 =21 6+1
8 =22 6+2
9 =23 6+3
10 =31 9+1
11 =32 9+2
12 =33 9+3
13 =111 9+3+1
14 =112 9+3+2
15 =113 9+3+3
16 =121 9+6+1
17 =122 9+6+2
18 =123 9+6+3
19 =131 9+9+1
20 =132 9+9+2
21 =133 9+9+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

Ternary maybe the best choice for checking out the results of your
generic recursive base implementation since it fairly easy to follow
what is goin on.
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

Counting is a collection of discrete entities ranging from first to
last member (inf
is not member of any set).
The first member in counting numbers in is generally one or 1,
unless
you do not adhere to some headless infinity working collective.
Below you can see sets? of discrete natural items and the summation
of
members that make up a set of countable naturals, as you see they
range from first to last since their countable and they are the
reason
numbers have comparable magnitudes, 1 is the base unit of math it
does
have a comparable magnitude, you can cut it to make fractions, count
it to make sets with comparable magnitudes. The whole idea of
numberline is wrong since 1 do not have any geometric properties/
attributes. It does have a magnitude though since it is divisible
into
fractions, the cuts from fractions also have magnitudes that
comparable to 1. Partitioning into bases is a principle with
geometric
properties, but base one have no other projection than counting from
the first to the last discrete member making up a natural number.

1={1}
2={1,1}
3={1,1,1}
4={1,1,1,1}
5={1,1,1,1,1}
6={1,1,1,1,1,1}
7={1,1,1,1,1,1,1}
8={1,1,1,1,1,1,1,1}
9={1,1,1,1,1,1,1,1,1}
A={1,1,1,1,1,1,1,1,1,1}

If you want attempt to make an algorithm that will encode
any decimal number with decimal parts into zeroless anybase, you can
use my standard base changer below.
It is maximum two lines of code.
http://www.anybase.co.nf/

JT

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Apr 15, 2013, 11:48:47 PM4/15/13

to

I hope you noticed that the first example was senary base 5 and the
latter base 3 ternary.

WT Shaw

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Apr 16, 2013, 1:25:07 AM4/16/13

to

I've been waiting to see if you guys would even extol the virtues of doing mathematics in Roman Numerals.

WTShaw

Peter Fairbrother

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Apr 16, 2013, 12:41:23 PM4/16/13

to

On 16/04/13 06:25, WT Shaw wrote:
> I've been waiting to see if you guys would even extol the virtues of doing mathematics in Roman Numerals.
>
> WTShaw
>

I want to calculate 3^verybignumber modulo a 6kbitnumber, many times.

Verybignumber is about 6 Megabits long and doesn't change often,
6kbitnumber changes each time.

Would trits help?

(the application is completely serious)

-- Peter Fairbrother

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