Natural numbers
JT
And not A=1 ten
Why do we write 20= two tens+zero
And not 2A=two tens
I want an explanation for the use of empty positions within natural
numbers, i guess this is a valid question for anybase?
JT
Maybe this is just sematic difference, but there must been an idea
behind the semantics.
Why is the zero there it sure must represent something like zero
hundreds, zero thousands and so on.
??????????????????????????????
They can hardly be there for the pupose of addition, or subtraction
and not for multiplication and not for division. So what is the zero
hundreds and zero thousands doing in the natural numbers?
JT
My feeling is that someone wanted 0 to be a natural number, is it a
natural number?
The romans did use a decimal number system but excluded 0.
Who included zero within the natural numbers?
Tim Golden BandTech.com
I do believe that there is room for criticism nearby to your
awareness.
Note that all number systems will be
base 10
number systems, so it is wise here to use instead
base ten
to imply our usual human digit based system, which is an arbitrary
choice.
The idea that the modulo numbers are more fundamental than the natural
numbers can be taken seriously, especially when for large enough n we
see no problem in adding or multiplying two other values m1 and m2.
Another important aspect that is largely overlooked is that
traditional sign (e.g. the real number, which has two signs) is
actually modulo two behaved with regard to its product. Note that we
normally distinguish between sum and product, but the sign product is
the modulo two sum where
- - = + , - + = - , + + = + .
There are some notational interpretations here, not unlike your own
modification of notation.
I think that for you to make sense you should identify some conflict
with the existing representaion, or point out how your own choice is
more coherent. Notational issues can be a matter of choice, but if
there are side effects to the choice which cause even a mild
difference in mechanics then that choice becomes more substantial.
- Tim
JT
wrote:
A number can be viewed as a representation of the sum of members with
different magnitude?
set(y)=(x*1,x*10,x*100,x*1000...)
Above is probably a total wrong notation and way to use set. But when
we write out magnitudes of different positions within the natural
number they ought to represent something right?
What i do not get is why 3001 would be written out.
set(y)=(1*1,0*10,0*100,3*1000)
when it can be written
set(y)=(1*1,3*1000)
But of course without zeroes... in 1000
Why are those zero magnitudes there inside the number. Do they have a
purpose?
Transfer Principle
> Why do we write 10=1 ten+zero
> And not A=1 ten
> Why do we write 20= two tens+zero
> And not 2A=two tens
According to Wikipedia, this can make sense. There exists a
nonstandard numeration system called "bijective numeration."
http://en.wikipedia.org/wiki/Bijective_numeration
"Bijective numeration is any numeral system that establishes
a bijection between the set of non-negative integers and the
set of finite strings over a finite set of digits. In
particular, bijective base-k numeration represents a
non-negative integer by using a string of digits from the
set {1, 2, ..., k} (k ≥ 1) to encode the integer's expansion
in powers of k.
"The bijective base-10 system is also known as decimal
without a zero. It is a base ten positional numeral system
which does not use a digit to represent zero; it instead has
a digit to represent ten, such as 'A'.
"As with conventional decimal, each digit position
represents a power of ten, so for example 123 is 'one
hundred, plus two tens, plus three units'. All positive
integers which are represented solely with non-zero digits
in conventional decimal (such as 123) have the same
representation in decimal without a zero. Those which use a
zero need to be rewritten, so for example 10 becomes A,
conventional 20 becomes 1A, conventional 100 becomes 9A,
conventional 101 becomes A1, conventional 302 becomes 2A2,
conventional 1000 becomes 99A, conventional 1110 becomes
AAA, conventional 2010 becomes 19AA, and so on."
Notice that twenty becomes 1A, not 2A or AA. For 1A has a 1
in the tens place and an A (ten) in the ones place, to make
ten plus ten, or twenty.
> I want an explanation for the use of empty positions within natural
> numbers, i guess this is a valid question for anybase?
It is valid for any base, including base 1. Indeed, the
Wikipedia link above explains this more fully.
In the bijective base-10 system, the current year would be
written as 1A11. There is a 1 in the thousands place, an A
(ten) in the hundreds place (i.e., two thousand so far), a
1 in the tens place and a 1 in the ones place. So, Happy
New Year 1A11, JT!
adamk
> wrote:
> > Why do we write 10=1 ten+zero
> > And not A=1 ten
> >
> > Why do we write 20= two tens+zero
> > And not 2A=two tens
> >
> > I want an explanation for the use of empty
> positions within natural
> > numbers, i guess this is a valid question for
> anybase?
>
> Maybe this is just sematic difference, but there must
> been an idea
> behind the semantics.
> Why is the zero there it sure must represent
> something like zero
> hundreds, zero thousands and so on.
>
> ??????????????????????????????
I would think the idea of including the zero is to
eliminate the ambiguity of the coefficient of units,
i.e., if we use base b, to make it clear that our
number is of the form
a_nb^n+ a_(n-1)b^(n-1)+...+a_1b^1+ 0b^0
and not that we may have forgotten to state the
coefficient of b_0.
It may be redundant in some sense, but maybe
redundancy is better than ambiguity. Or, maybe
as Oprah would put it, the zero on the right
gives you "closure" , i.e., you know the represe-
ntation is complete.
Maybe someone who does programming can tell you
the angle from the perspective of the way the computer
enters numbers into memory and/or the way the computer
adds (this clearly applies only to base 1, but I think
it generalizes.)
spudnik
whose canonical digital representation you can get
by induction on base-ten.
the real issue is that Maxwell flubbed
on Ampere's "longitudinal force,"
although it's a simple experiment ... haven't done it, doh!
M&M foound a regular, very small anomaly, which is almost
always refered to as a "null result" by the encyclopaedists (and
the googoloographers). what put it to rest was the development
of "atomism," namely electromagnetism, and it is almost
that simple.
Alfven probably put the Big Bang to bed but, again,
the encyclopaedists are about N decades behind the physics
of plasma, and *he* was also put finally to rest,
in the meantime.
the only real problem with special relativity is the slogans
about phase-space (spacetime), but Minkowski was one
to put his pants on, one lightcone at a time, two.
> put to bed by the Michelson-Morley experiment put this firmly
--GMMXI,not myIQ; yours? \
http://wlym.com
spudnik
JT
But if A = 10 and AA = 100 and this was a number system on its own
then would it not follow that 2A = 20, 9A = 90.
Or would that not work?
Where is the flaw in the use of 2A for 20?
The bijective system seem to say 90+10=100 my way of writing is more
spot on AA=100 the quirk is it seem to use powers of ten to express
positional magnitudes rather then addition as within the bijective.
But this may be explained by looking at a number as a set of
magnitudes, and then in will be all clear that 3001 written as a set
of positional magnitudes will look like this.
3001=set(y)=(1*1,3*AAA)
3902=set(y)=(2*1,9*AA,3*AAA)
3840=set(y)=(4*A,8*AA,3*AAA)
And as you can see within this way using powers i avoid zero, so what
do you think is it consistent?
JT
JT
JT
Maybe this is a bit like a radix based system in base 10?
JT
I forgot 100 and 1000 as sets of powers will of course be written.
100=set(y)=(AA) or (1*AA) but the 1 seem todo no good.
1000=set(y)=(AAA)
and from this follow of course
1001=set(y)=(1,AAA)
Ken Pledger
<ad726560-03e9-4f44...@fo10g2000vbb.googlegroups.com>,
JT <jonas.t...@hotmail.com> wrote:
Read Karl Menninger, "Number Words and Number Symbols - A Cultural
History of Numbers".
Ken Pledger.