"unary" neumber system
Brian Christiansen
the definition that a baseN system has the digits 0..N-1 available, then it
would only have the digits 0..0 available. A system that can only represent
the number 0 is pretty useless as far as I am concerned.
If "unary" simply means using tick marks to represent a number (1=/, 2=//,
3=///, etc., where / is a "tick mark"), then calling it unary seems to me it
is more sort of an informal designation rather than any sort of actual
mathematical numbering system.
What brought all this about is the other day I was at ace hardware and
purchased an item for 8.15, and the change back from the $10 was 1.85.
Another time, I got something at a mcdonalds that cost 2.03. So I did not
have to carry around 97 cents worth of change, I gave the cashier $5.05
instead of just a 5 dollar bill.
In both cases, the numbers that add up to the total have the same digits,
but in a different order (1000 = 815 + 185, 505 = 302+203). I was wondering
if there was an way to determine if a number meets that definition other
than by happening to notice it like I did or going through all the
possiblities:(1,N-1) (2,N-2), etc.
Then I got to thinking that the smallest such number would be 11=10 + 01,
and further got to thinking that, if you picked the appropriate base, any
number greater than 3 can be represented that way. The numbers 1 and 2
cannot be represented this way, or at least I don't think they can.
Even if you represent 2 as // in "unary" (if there is a numbering system
called "unary") or the "tick-mark-system" (if there is not), I dont think it
can be broken down into 11 = 10+01. For the number 1, there aren't even
two numbers that add up to it, at least not 2 positive numbers.
Brian Christiansen
Transfer Principle
<brian_christi...@hotmail.com> wrote:
> Is there any such thing as a "unary" (base 1) number system. If you go by
> the definition that a baseN system has the digits 0..N-1 available, then it
> would only have the digits 0..0 available. A system that can only represent
> the number 0 is pretty useless as far as I am concerned.
This Wikipedia link explains it all:
http://en.wikipedia.org/wiki/Unary_numeral_system
> If "unary" simply means using tick marks to represent a number (1=/, 2=//,
> 3=///, etc., where / is a "tick mark"), then calling it unary seems to me it
> is more sort of an informal designation rather than any sort of actual
> mathematical numbering system.
Yes, that's exactly what it is.
Tim Golden BandTech.com
Errr... I don't believe this.
Consider that in a base ten number system the value
345
is the same as
3 x 10 x 10 + 4 x 10 + 5 ,
the above being the radix ten interpretation of the symbolic value.
This is exactly consistent with a radix one interpretation of
1111
as
1 x 1 x 1 x 1 + 1 x 1 x 1 + 1 x 1 + 1
which I believe is consistent formally. The usage of 10^0 or 1^0 has
been avoided, and the existence of zero in the base one system does
seem worth consideration since it does not make its presence felt.
Typically we have 1 mod 1 = 0 for base one, which I believe is to say
that one and zero are the same thing in the base 1 numbers. For
instance the mod-2 numbers are regarded as having two elements:
0,1
but it is possible to restate this as
1,2
except that this conflicts with radix usage. This is nearly the same
as admitting that all radixed number systems used modulo 10 math,
where 10 is in that radix. There is a notational quandry here that
could be important.
Likewise sign can be generalized and the one-signed numbers pose a
seeming paradox, yet the paradox is consistent with that of time.
The modulo one interpretation in counting is the most fundamental form
so it deserves more respect from you two.
- Tim
James Dow Allen
news:2cc2c0f2-5ca5-4b51...@f30g2000yqa.googlegroups.com:
> This is exactly consistent with a radix one interpretation of
> 1111
> as
> 1 x 1 x 1 x 1 + 1 x 1 x 1 + 1 x 1 + 1
> which I believe is consistent formally.
> ...
> The modulo one interpretation in counting is the most fundamental form
> so it deserves more respect from you two.
FWIW, I agree with Tim. Those who insist "Stone-Age counting" (Glen
Langdon's term) shouldn't be called "unary" are overly pedantic, IMO.
Base-10 has 10 symbols, Base-2 has two; Base-1 has 1. The only
inconsistency is that you have to throw away zero instead of one when
you reduce the alphabet to a singleton: Otherwise you couldn't count
past zero. :-)
I'm also happy to speak of the Base-Phi (Fibonacci) counting system,
where the first few positive integers are
1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010
Base-Phi is sometimes used in compaction codes. Any other applications?
James Dow Allen
kunzmilan
"Tick marks" were found on bones. It was the first number system used
in the earliest times.
Now, we use it as the logarithmic base, as: // = 1^2.
kunzmilan
Henry
wrote:
> FWIW, I agree with Tim. Those who insist "Stone-Age counting"
> (Glen Langdon's term) shouldn't be called "unary" are overly
> pedantic, IMO.
> Base-10 has 10 symbols, Base-2 has two; Base-1 has 1. The only
> inconsistency is that you have to throw away zero instead of one
> when you reduce the alphabet to a singleton: Otherwise you
> couldn't count past zero. :-)
You can also have Base-n systems with the n symbols 1, 2, ... n but
not zero for larger values of n. See http://en.wikipedia.org/wiki/Bijective_numeration
Brian Christiansen
"Brian Christiansen" <brian_ch...@hotmail.com> wrote in message
news:ijciqn$4b1$1...@news.eternal-september.org...
All very interesting, but I was actually more interested in the following:
> What brought all this about is the other day I was at ace hardware and
> purchased an item for 8.15, and the change back from the $10 was 1.85.
> Another time, I got something at a mcdonalds that cost 2.03. So I did not
> have to carry around 97 cents worth of change, I gave the cashier $5.05
> instead of just a 5 dollar bill.
>
> In both cases, the numbers that add up to the total have the same digits,
> but in a different order (1000 = 815 + 185, 505 = 302+203). I was
> wondering if there was an way to determine if a number meets that
> definition other than by happening to notice it like I did or going
> through all the possiblities:(1,N-1) (2,N-2), etc., or specifically
> constructing thed number that way (if limiting yourself to base10 becausee
> any number can be expressed as 11=10+01, at least any number larger than
> 3)
This is what happened:
1. I noticed the stuff about the change.
2. Got to wondering if there was a way to tell if there was a way to tell
if a number met that criterion without happening to notice it, generating
all the possibilities, or specifically constructing it that way. This is my
primary question.
3 Got to thinking that the smallest such number is 11 (10+01), and that if
you pick the appropriate base, any number 3 or larger can be expressed that
way, but I am not so sure about the number 2 (pretty sure it cannot, but not
100%). This got me off on the tangent (at least I consider it a tangent to
my primary question) about the unary numbering system.
Brian Christiansen.
James Waldby
> [...] I was at ace hardware and
> purchased an item for 8.15, and the change back from the $10 was 1.85.
> Another time, I got something at a mcdonalds that cost 2.03. So I did
> not have to carry around 97 cents worth of change, I gave the cashier
> $5.05 instead of just a 5 dollar bill.
>
> In both cases, the numbers that add up to the total have the same
> digits, but in a different order (1000 = 815 + 185, 505 = 302+203). I
> was wondering if there was an way to determine if a number meets that
> definition other than by happening to notice it like I did or going
> through all the possibilities:(1,N-1) (2,N-2), etc.
...
I don't know of a formula or an elegant method for writing n as the
sum of two numbers u, v where the digits of u match those of v.
However, the method given below is faster than the O(n) method you
mentioned. In following, I suppose 1000 > u >= v > 0. This method
is extensible to larger n but I don't know if it remains O(1).
Suppose u = 100a + 10b + c. We seek an order p,q,r of a,b,c such that
v = 100p + 10q + r and n = u+v; or (*) n = 100a+10b+c + 100p+10q+r.
The possible orders pqr are abc, acb, bac, bca, cab, and cba.
(*) is as follows for the different orders:
abc: n=200a+20b+2c. acb: n=200a+11(b+c). bac: n=110(a+b)+2c.
bca: n=101a+110b+11c. cab: n=110a+11b+101c. cba: n=101(a+b)+20b.
Each of these linear Diophantine equations can be solved in constant
time per solution (with various gcd's precomputed).
--
jiw
David R Tribble
>> This is exactly consistent with a radix one interpretation of
>> 1111
>> as
>> 1 x 1 x 1 x 1 + 1 x 1 x 1 + 1 x 1 + 1
>> which I believe is consistent formally.
>> ...
>> The modulo one interpretation in counting is the most fundamental form
>> so it deserves more respect from you two.
>
James Dow Allen wrote:
> FWIW, I agree with Tim. Those who insist "Stone-Age counting" (Glen
> Langdon's term) shouldn't be called "unary" are overly pedantic, IMO.
> Base-10 has 10 symbols, Base-2 has two; Base-1 has 1. The only
> inconsistency is that you have to throw away zero instead of one when
> you reduce the alphabet to a singleton: Otherwise you couldn't count
> past zero. :-)
Or even to zero, since it's a complete absence of tick marks.
> I'm also happy to speak of the Base-Phi (Fibonacci) counting system,
> where the first few positive integers are
> 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010
> Base-Phi is sometimes used in compaction codes. Any other applications?
See Knuth's quater-imaginary number system, which uses
a complex base of 2i:
http://en.wikipedia.org/wiki/Quater-imaginary_base
rasterspace
for base-one, by induction on base-ten.
Rock Brentwood
<brian_christi...@hotmail.com> wrote:
> Is there any such thing as a "unary" (base 1) number system. If you go by
> the definition that a baseN system has the digits 0..N-1 available, then it
> would only have the digits 0..0 available. A system that can only represent
> the number 0 is pretty useless as far as I am concerned.
Well, first: numbers are not unary or anything-ary. Ary-ness is a
property of the *numerals* used to represent numbers, not the numbers
themselves.
Numerals are orthographic systems. So technically, this is not a
quwestion of mathematics at all, but a question of Linguistics (both
syntax and semantics); since orthography is a part of Linguistics.
Second, the property generalizes to systems of numerals to represent
ANY linear associative algebra (e.g. complex numbers, quaternions,
Clifford algebras). For base N, one requires a *unique* decomposition
of each element a of the underlying set A into the form a_0 + a_1 N +
a_2 N^2 + ...; where the digits {a_0, a_1, a_2, ...} are taken out of
a fixed set D. In order to handle algebras over the field of real
numbers one can extend this to include negative powers a_{-1} N^{-1} +
a_{-2} N^{-2} + ...; then the condition is that the numerals of the
form:
sum_{n=-K}^{L} a_n N^n; for K, L non-negative integers
yields a denose subset of the algebra A.
Numerals for complex numbers in base 3 include D = {-1-i, -i, -1+i,
-1, 0, 1, -1+i, i, 1+i} which could be denoted by {L, U, J, C, O, D,
F, A, 7}. Also you can have base 3i with the numerals D = {-4, -3, -2,
-1, 0, 1, 2, 3, 4}. This also works to give you the integers with base
9. Base 3 integers can also be written with D = {-1, 0, 1} -- the
smallest base that works with all integers (not just the non-negative
ones).
Quaternions can be represented with numerals in base 3 with 81 digits
to denote the values of {w + xi + yj + zk} as {w,x,y,z} range over
{-1,0,1}.
In all these orthographic systems, the usual algorithms for +, -, *
and / apply, with minor modifications or extensions. Using the iconic
digits for the base 3i complex numbers, multiplication by i is
repersented by rotating the symbol 90 degrees counter clockwise. (So
you should actually use the rotational variants of L in place of 7, F
and J; the variants of U in place of C, D and A).
Going by this definition, base 1 uses powers of 1 for N and requires
unique representation. Uniqueness can be fudged since the order of the
digits no longer matters. The set D = {1} will work to represent non-
negative integers.
So, base 1 is the smallest base for non-negative integers; base 2 is
the smallest base for natural numbers {0,1,2,3,...}, and base 3 is the
smallest base for the integers (or reals, complex numbers, quaternions
or any other finite dimensional real linear algebra, including all
finite dimensional matrix algebras).
So, as an exercise: devise a system of numerals to denote the elements
of 2x2 matrices in base 3 or 9.
Rock Brentwood
>
> So, as an exercise: devise a system of numerals to denote the elements
> of 2x2 matrices in base 3 or 9.
As a diversion: one can also generalise the convention IV = V - I to
come up with a sophisticated algebraic/combinatorial system of
numerals for the inteters. The rules posed for the "base form" of the
numerals is that numerals consists of a sequence of digits
(I,V,X,L,C,D,M) interspersed by commans and dashes. The values of the
letter digits are I=1, V=5, X=10, L=50, C=100, D=500, M=1000; the
value denoted by a sequence of letter digits IN NON-INCREASING ORDER
is the sum of the values of each digit, and the rules for handling the
dashes and commas are:
(1) a dash means multiply everything to its left by 10
(2) a comma means multiply everything to its left by 1000
The rules apply iteratives (e.g. A,B,C = 1000000 A + 1000 B + C if A,
B, C are letter sequences.
If the digits are not in non-increasing order, then a generalization
of the IV rule applies, as described here. Let S be the letter
sequence. Take the unique decomposition S = S0 d0 S1 d1 ... Sn dn
where d0, d1, ..., dn are a non-increasing sequence of digits and S0,
S1, ..., Sn are (possibly empty) sequences of digits smaller than the
d-digits that bound it. Then the value of S is
the value of (d0 d1 d2 ...) minus the sum of the values of S0,
S1, ..., Sn.
Thus, for instance, VVIX = X - VVI = -1; while
IXVILDC = DC - (IXVIL)
= DC - (L - (IXVI))
= DC - (L - (XVI - I))
= DCXVI - LI
= DLXV
= 565.
There is a simple, elegant combinatorial algorithm for addition, and a
second combinatorial algorithm which is equally elegant for
multiplication. I'll leave their specifications as exercises.
rasterspace
it has to be a delimited "space."
David Bernier
> On Feb 14, 6:53 pm, "Brian Christiansen"
> <brian_christi...@hotmail.com> wrote:
>> Is there any such thing as a "unary" (base 1) number system. If you go by
>> the definition that a baseN system has the digits 0..N-1 available, then it
>> would only have the digits 0..0 available. A system that can only represent
>> the number 0 is pretty useless as far as I am concerned.
>
> Well, first: numbers are not unary or anything-ary. Ary-ness is a
> property of the *numerals* used to represent numbers, not the numbers
> themselves.
>
> Numerals are orthographic systems. So technically, this is not a
> quwestion of mathematics at all, but a question of Linguistics (both
> syntax and semantics); since orthography is a part of Linguistics.
>
> Second, the property generalizes to systems of numerals to represent
> ANY linear associative algebra (e.g. complex numbers, quaternions,
> Clifford algebras). For base N, one requires a *unique* decomposition
> of each element a of the underlying set A into the form a_0 + a_1 N +
> a_2 N^2 + ...; where the digits {a_0, a_1, a_2, ...} are taken out of
> a fixed set D. In order to handle algebras over the field of real
> numbers one can extend this to include negative powers a_{-1} N^{-1} +
> a_{-2} N^{-2} + ...; then the condition is that the numerals of the
> form:
> sum_{n=-K}^{L} a_n N^n; for K, L non-negative integers
> yields a denose subset of the algebra A.
What do you mean by a "denose subset" of the algebra A?
David Bernier
--
"isn't quotable" isn't quotable.
James Waldby
> Rock Brentwood wrote:
...
>> Numerals are orthographic systems. So technically, this is not a
>> quwestion of mathematics at all, but a question of Linguistics (both
>> syntax and semantics); since orthography is a part of Linguistics.
>>
>> Second, the property generalizes to systems of numerals to represent
>> ANY linear associative algebra (e.g. complex numbers, quaternions,
>> Clifford algebras). For base N, one requires a *unique* decomposition
>> of each element a of the underlying set A into the form a_0 + a_1 N +
>> a_2 N^2 + ...; where the digits {a_0, a_1, a_2, ...} are taken out of a
>> fixed set D. In order to handle algebras over the field of real numbers
>> one can extend this to include negative powers a_{-1} N^{-1} + a_{-2}
>> N^{-2} + ...; then the condition is that the numerals of the form:
>> sum_{n=-K}^{L} a_n N^n; for K, L non-negative integers
>> yields a denose subset of the algebra A.
> [...]
>
> What do you mean by a "denose subset" of the algebra A?
Probably that's just a typo for "denosed subset", ie, one
that has been deprived of its sense of smell.
--
jiw
Ilmari Karonen
>
> So, base 1 is the smallest base for non-negative integers;
Excluding zero from the set of base-1 numbers seems a bit arbitrary,
since it has a perfectly valid representation as "", i.e. an empty
string of digits.
Of course, in number systems which include the digit 0, it's
conventional to use the string "0" to represent zero instead of "",
but this is really an exception to the rule that canonical
representations of numbers should not have leading 0 digits.
Of course, this exception is quite convenient, since it allows us to
distinguish zero from the absence of a number without the need for
delimiters or other external cues, but it is not strictly necessary.
After all, people have been using tally marks (a base-1 number system)
to represent integer quantities, including zero quantities, for ages.
> base 2 is the smallest base for natural numbers {0,1,2,3,...}, and
> base 3 is the smallest base for the integers (or reals, complex
> numbers, quaternions or any other finite dimensional real linear
> algebra, including all finite dimensional matrix algebras).
Well, depends on what you mean by smallest, I guess. On can represent
the integers (and reals, with negative powers) using the digits 0 and
1 and the base -2. Or even all the complex numbers using the digits 0
and 1 and base sqrt(2)i (again with negative powers).
In fact, the complex numbers can be "uniquely" (see below) represented
in base 2 using the digits {1, -1, i, -i}. Of course, this means that
zero must be represented either as the empty digit string or as a
non-terminating digit string of the form 1.(-1)(-1)(-1)... (or
(-1).111... or i.(-i)(-i)(-i)... or (-i).iii...).
Also, as the above shows, non-terminating representations are not
always unique in this system -- but then, the same is true of your
base-3 representation of C (where 0.111... = 1/2 = 1.(-1)(-1)(-1)...,
and where (i+1)/2 even has four representations), and even of the
traditional decimal representation of R^+. (Cue yet another long
thread about whether 0.999... *really* equals 1.) The best we can
really ask for is that the terminating representations be unique and
dense in the set being represented, which this system satisfies.
--
Ilmari Karonen
To reply by e-mail, please replace ".invalid" with ".net" in address.
Ilmari Karonen
> On 2011-02-16, Rock Brentwood <federat...@netzero.com> wrote:
>>
>> So, base 1 is the smallest base for non-negative integers;
>
> Excluding zero from the set of base-1 numbers seems a bit arbitrary,
> since it has a perfectly valid representation as "", i.e. an empty
> string of digits.
Mmm... also, "non-negative" normally means *including* zero, although
I assume(d) from context that you meant to exclude it here, since you
then went on to write:
>> base 2 is the smallest base for natural numbers {0,1,2,3,...}, and
>> base 3 is the smallest base for the integers (or reals, complex
>> numbers, quaternions or any other finite dimensional real linear
>> algebra, including all finite dimensional matrix algebras).
--
jbriggs444
> zero can be represented in base-one, but
> it has to be a delimited "space."
Of course. The empty string encodes zero.
I think that what you are trying to point out that it is
difficult to distinguish between one stretch of white space
and two stretches of white space with a null string in
between. Which, of course, is more a matter of typography
than of mathematics.
Our conventions for leading zeroes are similarly more a
matter of convention and typography than of mathematics.
David Bernier
> On Tue, 15 Feb 2011 23:58:01 -0500, David Bernier wrote:
>> Rock Brentwood wrote:
> ...
>>> Numerals are orthographic systems. So technically, this is not a
>>> yields a denose subset of the algebra A.
>> [...]
>>
>> What do you mean by a "denose subset" of the algebra A?
>
> Probably that's just a typo for "denosed subset", ie, one
> that has been deprived of its sense of smell.
Thanks! I was wondering if you might have pointers to the
Google query syntax. For example, to assert no fuzzy guessing,
it seems one has to precede a query word by the plus sign, as:
"+Haros" for instance (to avoid triggering Haro<-->Haros, etc.),
or minus sign to exclude a query word, e.g. "-facebook" , etc.
James Waldby
The clearest information I've seen is at <http://www.googleguide.com/> on
pages such as <http://www.googleguide.com/interpreting_queries.html> and
<http://www.googleguide.com/using_advanced_operators.html>. + marks a
word as required and also turns off automatic stemming; double quotes
around a phrase do the same for all words of the phrase. * stands for
any word. ~ turns on synonyms.
--
jiw
rasterspace
did anyone get the canonical digital representatiion
of base-one?
:Scoll ye God-am palimpsesT:
"writing numbers backwards" was the problem
with Hensel's lemma for p-adic numbers;
there is really no sufficient reason to do that.
also, see Laurent sreies.
:Scoll ye God-am palimpsesT:
there is nothing weird about space being "curved,"
if interepreted in terms of refraction. however,
Minkowski's useless spacetime slogan seems
to have been used by the British Pyshcological Research Soc.
to create a popular misconception, known as Flatland.
and Minkowski was a great "N-d" geometer,
viz his generalization of Pick's theorem.
:Scoll ye God-am palimpsesT:
to say that Newton's corpuscle was
in any way a classical theory, belies the fact that
it was not a theory, at all, considering that
he got refraction completely wtong. which is hard to do,
if he really answered Liebniz's challenge
on the brachostochrone -- supposedly *the* classical problem
to initiate the caclulus by Bernoulli et al.
Tim Golden BandTech.com
> On 2011-02-16, Rock Brentwood <federation2...@netzero.com> wrote:
>
>
>
> > So, base 1 is the smallest base for non-negative integers;
>
> Excluding zero from the set of base-1 numbers seems a bit arbitrary,
> since it has a perfectly valid representation as "", i.e. an empty
> string of digits.
>
> Of course, in number systems which include the digit 0, it's
> conventional to use the string "0" to represent zero instead of "",
> but this is really an exception to the rule that canonical
> representations of numbers should not have leading 0 digits.
This is worth some careful consideration I believe. That zero and one
can have the same meaning within the modulo one numbers may seem a
flaw, but is the flaw actually one of our notational conventions?
Traditionally if we treat the natural numbers
1, 2, 3, 4, ...
as fundamental then we can construct the modulo two form of the
natural numbers as
1, 0, 1, 0, ...
but under this standard we see that the modulo one numbers are
0, 0, 0, 0, ...
and that the meaning of this is just the same as
1, 1, 1, 1, ...
which is to say that under the modulo one numbers
1 = 0 .
This is fairly apparent within polysign mathematics, where this modulo
one behavior is taken by the sign of the one-signed numbers P1.
Notationally it is useful to introduce the zero sign because it is
universal and takes on what in traditional modulo notation is a moving
beast; the highest value, whose handicap exposes the notational
failure of the standard modulo system: when we speak of modulo 2
numbers the element 2 does not exist, which can be regarded as a
conflict. When we get to a radix notation all number systems become
'radix 10' within their standard notation, but for the modulo one
numbers this rule does not fit.
Resolution of this notational failure could be a choice of whether the
natural numbers are
0, 1, 2, 3, ...
or
1, 2, 3, 4, ...
but it seems a bit more complicated than that. Most importantly it is
possible to build these low modulo sets without ever defining the
natural numbers, and since simpler things should not be built from
more complex things then this is a criticism on the entire
nomenclature of 'modulo' arithmetic.
The symbolic choice of the elements of the natural numbers is
arbitrary, yet the existence of unity is not, and that is the critical
stepping element.
Another key feature of the modulo one numbers is that their
superposition is their summation, and this only works with their
domain:
111 + 11 = 11111
Isn't it true that this modulo one representation is ultimately what
all of the radix interpretations mean? Isn't this the core? The symbol
2 in its most fundamental form is
1 1, or . ., or 0 0;
these symbols above are arbitrary; they must be so. The primitive
nature of the modulo one representation may seem beneath us, but their
fundamental status is worth troubling over.
- Tim
rasterspace
you almost got the canonicals.
Tim Golden BandTech.com
> wow & sad, if you mean base-one numbers ... but
> you almost got the canonicals.
Yeah, sorry I misuse 'modulo-one' to mean base one, but then there are
these notational issues that creep in.
- Tim
rasterspace
JT
> On 2011-02-16, Rock Brentwood <federation2...@netzero.com> wrote:
>
>
>
> > So, base 1 is the smallest base for non-negative integers;
Yes it is
> Excluding zero from the set of base-1 numbers seems a bit arbitrary,
No arbitrary is to include 0 to begin with since it not a numerical
operand.
> since it has a perfectly valid representation as "", i.e. an empty
> string of digits.
The idea of a base having range -1 is bad arithmetic to start with
base 16 is really 1-16 not 0-15 it is bad , in programming you have to
make dirty patches to make up for the flaws of mathematic included in
programming languages.
> Of course, in number systems which include the digit 0, it's
> conventional to use the string "0" to represent zero instead of "",
> but this is really an exception to the rule that canonical
> representations of numbers should not have leading 0 digits.
> Of course, this exception is quite convenient, since it allows us to
> distinguish zero from the absence of a number without the need for
> delimiters or other external cues, but it is not strictly necessary.
> After all, people have been using tally marks (a base-1 number system)
> to represent integer quantities, including zero quantities, for ages.
We have nothing,nada, null and NaN already the idea behind 0 as a
natural is ludicrous to start with. 0 have no dimension, no magnitude
no size it is not a number.
rasterspace
you're IQ is at least zero,
kind of a lower bound.... seriously,
what is the canonical digital representation
for unary, by induction on base-ten?... well,
you ccan skip a few steps.
Ostap Bender
Sure.
I is one
II is two
III is three
and so on.
Homework: write down the number 1 000 000 in this notation.
> If you go by
> the definition that a baseN system has the digits 0..N-1 available, then it
> would only have the digits 0..0 available. A system that can only represent
> the number 0 is pretty useless as far as I am concerned.
>
> If "unary" simply means using tick marks to represent a number (1=/, 2=//,
> 3=///, etc., where / is a "tick mark"), then calling it unary seems to me it
> is more sort of an informal designation rather than any sort of actual
> mathematical numbering system.
>
> What brought all this about is the other day I was at ace hardware and
> purchased an item for 8.15, and the change back from the $10 was 1.85.
> Another time, I got something at a mcdonalds that cost 2.03. So I did not
> have to carry around 97 cents worth of change, I gave the cashier $5.05
> instead of just a 5 dollar bill.
>
> In both cases, the numbers that add up to the total have the same digits,
> but in a different order (1000 = 815 + 185, 505 = 302+203). I was wondering
> if there was an way to determine if a number meets that definition other
> than by happening to notice it like I did or going through all the
> possiblities:(1,N-1) (2,N-2), etc.
>
> Then I got to thinking that the smallest such number would be 11=10 + 01,
> and further got to thinking that, if you picked the appropriate base, any
> number greater than 3 can be represented that way. The numbers 1 and 2
> cannot be represented this way, or at least I don't think they can.
>
> Even if you represent 2 as // in "unary" (if there is a numbering system
> called "unary") or the "tick-mark-system" (if there is not), I dont think it
> can be broken down into 11 = 10+01. For the number 1, there aren't even
> two numbers that add up to it, at least not 2 positive numbers.
>
> Brian Christiansen
Tim Golden BandTech.com
> ... and I meant, in spite of polysignosis.
Well that is creepy.
There are some formal conflicts down in this region that are not
directly tied to polysign, but which do show themselves in polysign at
a very challenging position: the one-signed numbers, whose
representation matches that of time. The unidirectional and zero
dimensional nature of P1 is the modulo one number as sign married with
a continuous magnitude. There, the compound representation normally
indicates a product, as for instance in the reals
(-)(-)(+) = +
is the modulo two sum, where the juxtaposition of symbols implies
product, whereas in one-signed numbers we have
(-)(-)(-)(-) = -
so that the sign product is the modulo one sum.
Because we are so near to the symbolic level the notation itself has
to come into focus. All of our expressions that fit upon a line are in
a sense appended expressions whose breakdown is of an arbitrarily
imposed nature.
That there is some room down here for interpretation is promising.
What if we are overlooking an important option? If it is so simple as
to be overlooked then all the better. Perhaps we are getting caught up
in our string format. Perhaps there is room for a branch. That which
appears foreign and unintelligible may have utility. Unfortunately
when it is found it will appear foreign and unintelligible and may not
propagate well.
- Tim
FredJeffries
> the definition that a baseN system has the digits 0..N-1 available, then it
> would only have the digits 0..0 available. A system that can only represent
> the number 0 is pretty useless as far as I am concerned.
>
> If "unary" simply means using tick marks to represent a number (1=/, 2=//,
> 3=///, etc., where / is a "tick mark"), then calling it unary seems to me it
> is more sort of an informal designation rather than any sort of actual
> mathematical numbering system.
Semi-seriously, this discussion reminded me of the field with one
element:
http://en.wikipedia.org/wiki/Field_with_one_element
Seems to me that some clever person ought to be able to link the two.
After all, sets are vector spaces over the field with one element,
natural numbers count finite sets and the base n representations can
be considered a sort-of-module (with carry) over the cyclic ring with
n elements, ...
Well, maybe not.
rasterspace
which "one can get" by induction of base-ten, althogh
it's a kind of obviousness.
rasterspace
Ostap Bender
>
> "tick marks" are not the cononical representation,
Yes, you should have a doctor or a nurse examine them.
Tim Golden BandTech.com
Nice observations.
Another thought: The unary form does not allow rational
representations. For instance in binary notation we could have
1010.000101
no different than we might write
12.015
in 'decimal' notation. The option to split unity does not exist within
the unary numbers. The unaries are in stronger correspondence with the
natural numbers under this awareness. If we try
1.111
it is equivalent to
1111 .
I wouldn't be surprised if some readers are in denial of the
legitimacy of a modulo one mechanism. Time obeys similar properties
when these modulo systems are taken as sign. This is an overlooked
area in mathematics.
By fairly direct analogy we could consider the simplex in its general
n form. We can even give this simplex an additional position: its
central point. This happen to form the sign vectors of an n-signed
space, where their geometry becomes clear. Now as we descend from say
a 4 verticed simplex (tetrahedron) down to a three (equilateral
triangle) down to a two(double segment) and finally one, then what
about the center? Here our minds could be playing tricks upon us, for
we have been bred in a culture that treats the real number as
fundamental. That time exists; that time has unidirectional behavior;
that we are caught in the present with no freedom to alter our
position either toward the past or toward the future; These behaviors
carry correspondence with this one verticed simplex, especially under
polysign, where a continuous magnitude is married to the discrete
sign.
It is a bit askance to the topic here, but definitely related. The
choice to declare the unary invalid is not the general choice, which
operates on unlimited n. None the less we see in the human a tendency
to eliminate this unary feature, and likewise to puzzle over the
qualities of time.
There seems to be so little to shuffle down at this level, and such a
shuffle would upset the entire accumulation I think. If we wish to
work in the general mod-n consistent system then we are forced to work
in the natural numbers, since the ability to build rational values at
mod-1 is nonexistent. I find this area fascinating but difficult to
pry open. I admit I am drawn toward declaring the ambiguity now, and
this double speak deserves to be put out of its misery. As we strive
for unification; under its assumption we are still missing
something...
The tatrix and its spacetime correspondence propose that all of these
modulo systems exist in parallel and that our commitment to one of
them might be a misnomer. If when we discuss base ten values we ought
to have in mind all of the lesser bases as well- this is somewhat the
cryptic thought.
There are other arguments that I have tried to make on this topic in
the past and often the unary numbers can be practiced with 'marbles in
a bag' and nearly no mathematical expertise. This is a very desirable
situation and can lead to a mixed modulo system by iterating out the
procedure, and the arbitrary nature of the representation admits that
a standardized pattern could be founded. These mixed modulo
representations are possibly going to provide something unseen. This
is nearby to the Chinese remainder theorem but the dance of these
forms is clear without that level of number crunching. I will try to
put some time into these. Natural correspondence is already exposed;
what if there is even more?
- Tim
rasterspace
canonical representation of base-one,
by induction on a larger base.
Tim Golden BandTech.com
Base1 : Base2 : Base3 : Base4 : ...
1 : 1 : 1 ...
11 : 10 : 2 : 2 ...
111 : 11 : 10 : 3 : 3 ...
1111 : 100 : 11 : 10 : 4 : 4 : 4 ...
11111 : 101 : 12 : 11 : 10 : 5 : 5 ...
111111 : 110 : 20 : 12 : 11 : 10 : 6 : 6 : ...
1111111 : 111 : 21 : 13 : 12 : 11 : 10 : 7 : 7 ...
et cetera...
You can count on it...
The idea that a larger base can be taken as a more primitive principle
is flawed, or at least this would be the standard interpretation. We
do not need three to count to two. If any of these representations can
take fundamental status it is the unary, whose symbolism is
arbitrary.
To come from the other way suggests taking BaseInfinity numbers as
fundamental. These seem to take an easy correspondence to Base1, and
this way of thinking is interesting, and is even extensible to a
dimensional interpretation, which I've tried, but the argument on
simplicity coming first seems difficult to challenge.
The above two interpretations can only be trumped by a third: each of
the BaseN interpretations stand freely. To what degree then does each
deserve its own unique symbolism? This is a notational issue that
tends to confound polysign as well, where the meaning of '+' is
shifty, though some will map the f to a t.
- Tim
rasterspace
read one. I'm sure that others have already gotten it.
JT
wrote:
Homework write 0 in this notation
rasterspace
y'know, "no counters." but,
the canonical digit is certainly not a tickmark ...
by induction, kids.