Re: The common usage of "nonnegative real number" is ludicrous.

On Mar 24, 2:00 pm, "VK" <schools_r...@xxxxxxxxx> wrote:
On Mar 22, 3:24 am, "Timothy Golden BandTechnology.com"

<tttppp...@xxxxxxxxx> wrote:
Yes, and there are magnitudes also which are signless numbers that are
not necessarily zero, and that is usually what is meant by
"nonnegative real number". This is a simpler concept being defined by
a complicated construction. That is an ambiguity in existing
mathematics.

I don't see any ambiguity here.

The ambiguity is not direct. It is in the structural integrity of the
construction.
Certainly the traditional real number is self-consistent. However it
prevents its own generalization by denying its actual form. This also
exposes itself when we see magnitude (a simpler concept) being defined
as 'non-negative real numbers'. Here a simpler concept is defined by a
more complicated concept and I argue that the simpler concept was
already used to build the more complicated one. This is an ambiguous
construction. When I want to talk about magnitude I am talking about
something beneath the real number, but to you it is above the real
number because the real number is taken as fundamental by you.
Answering the question 'is magnitude simpler than the real number?'
will suffice to get the construction cleaned up.


Within the current abstraction schema there are negative numbers,
positive numbers and the sign change point which is 0. Respectively we
have
1) "non-negative numbers" so zero inclusive
2) "positive numbers" so zero exclusive
3) "non-positive numbers" so zero inclusive
4) "negative numbers" so zero exclusive

0 as being the sign change point doesn't have sign by its own. To make
it closer to some real life experience one can consider the
International Date Line. This line is located in the area of 180th
meridian - I'm saying "in the area" because the line doesn't strictly
follow the meridian but intentionally fancy curved to avoid any areas
with noticeable population. From one side of this line is say Friday,
from the other side it is Saturday, one day difference. When crossing
this line all ships are obligated to change the clocks and calendars
appropriately. Now it is possible to imagine two ships by opposite
sides of the International Date Line. They can be in view of each
other and still it is officially Friday on one ship and Saturday on
the other ship. It is even possible to imagine a bored captain who
located his ship with such coordinate precision what officially it
will be Friday on the bow and Saturday on the stern. But no one is
asking what day - Friday or Saturday - is "strictly exactly on the
International Date Line itself". This imaginary line defines a quantum
property change point and as such it does not have this property.

I like your date line construction, but that does also prompt me to
consider the continuous symmetry of the situation and query why one
singular point (or meridian) should matter so much as to affect the
language. That special point is zero and it happens to be where sign
changes. Your meridian model will lend itself to be applied accross
the entirety of the structure of its values whereas the real line has
proclaimed a special location. Quibbling over this sort of thing is
not really what I want to be doing. There is one particular interval
that gets used regularly:
x > = 0 .
This particular interval is of interest because when we admit that the
real number uses the
s x
notation (where s is sign and x is magnitude) we are merely asking
that the interval be the x portion of the real number. So we have
defined a simpler concept in the guise of a more complicated concept.
This is a structural ambiguity. If the real number is indeed composed
of
s x
then we should really be using the quality of x as the nonnegative
real numbers. This x portion is what built the real numbers so the
complicated route should be avoided.


If there were consequences to the statement
- 0 = + 0 = 0
then there might be room for concern, but the insistence that zero is
unsigned seems to carry no consequential difference than allowing it
to also carry a sign, and indeed it must be allowed to be generated
this way for if I introduce a value
a
and then express
- a
and then I allow a to be zero we will be forced to evaluate minus
zero. So all of this worry about zero is silly. Whether the positive
reals contain zero or not is just convention.

Yes, and one may say that strictly exactly on the International Date
Line itself is always Monday no matter what day of week is left or
right. Practically it has zero importance though looks rather silly.
The distinction of non-negative and positive numbers is not so much in
sign issues - if it is at all. It is a matter of considerations
because of some current arithmetical conventions. Say the equation x/
10 = x has no solution for positive natural numbers but it has one
solution for non-negative natural numbers.

All of this worry over
zero is tangential to the concept I am trying to focus on; the
symbollic usage of sign and the ignorance of the advanced
mathematician of the fundamental symbolic structure of his real
number. Such symbolism is not arbitrary and upon separating the
symbolic types we see two simpler types:
s x
where s is sign and x is magnitude.

IMHO that is too much of IEEE floating point number thinking here. A
negative number is not a number "with the sign bit set"; it is neither
a number with minus sign placed in front of it. A negative number is
just that: a number located from one side of the sign change point 0.
If you want a positive number, you don't "remove sign" - you are going
to the other side of the sign change point for all another number. So
the modularity you are proposing is not really suitable here.
If keeping s,e,m (sign, exponent, mantissa) or the proposed more
simple s,m (sign, magnitude) structure the number becomes some meccano-
like structure where everything is by itself and can be taken out
separately. "1 is 0 with magnitude and positive sign added", "-1 is 0
with magnitude and negative sign added" ? ;-)

This is not the IEEE standard we are talking about. This is every
concrete instance of the real number we are discussing here. You are
in denial that the symbology of your construction takes the form
s x
and so are fodder for my argument. For every value in the reals that
you pass a sign of that value can be expressed, and a magnitude. The
zero instance is trivial and we will all readily admit that
- 0 = + 0 = 0 .
This is a point of connection between the signatures of the system.
Nicely when this is expanded to three-signed values the graphical
arbitrariness of the zero point goes away since we then are dealing
with three unique rays drawn from the origin. The intersection is
fully exposed and so its special nature becomes more apparent than on
the unmarked real line.


Actually such "number modularity" is in effect on IEEE systems where
each number is just a bits sequence with different parts having
different purpose. It is interesting that in this case the difference
between -0 and +0 is implied: because these numbers have different bit
sets. It also implies the difference between operations with -0 and
0 : because of the existing arithmetical conventions we have to
account sign preservations and transformations. Some systems go the
full way down so there -0 != 0, some systems stops half way. For any
JavaScript-enabled browser one may "enjoy" for instance by this:

<html>
<head>
<title>Demo</title>
<script type="text/javascript">
var a = [-1, -0, 0, 1];
for (var i=0; i<a.length; i++) {
window.alert(1/a[i]);}

/* This outputs:
* -1
* -Infinity
* Infinity
* 1
*/
</script>
</head>
<body>
<p>Demo</p>
</body>
</html>

These types are more fundamental
than the real number which they create. There are consequences to this
which are not trivial yet this construction denies the mainstream
definition of the reals for when sign is generalized
Sum over s ( s x ) = 0
becomes fundamental. This rewrites the definition of the real numbers
and requires magnitude to be fundamental.

Having two rabbits one from left and one from right: the left rabbit
is not the right rabbit "with rightness property cleared and leftness
property set". These are two different rabbits by their own. Again:
IMHO you are trying to re-interpret numbers from a point of view of a
discrete storage system which is not really suitable.

P.S. From a linguistic point of view it is always possible to discuss
if "non-negative" English term is the best out of all possible.

Above here you get into negative infinity and I do take some interest
in that.
If as you say zero is unsigned then should negative infinity be
banished?
There is a tight relationship between these two concepts and the
denial that zero takes any sign whatsoever may even deny the
construction of a positive infinity. Here are instances where a signed
zero does matter. To some this might be seen as a tangential argument
but perhaps for others it is central. The ability to reciprocate
requires a signed zero if one wishes to make use of negative infinity
or positive infinity.

-Tim


.

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