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November 14th, 2013, 10:21 PM   #1
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Zero-infinity number arrangement based on number size.

Hello ladies and gentlemen, MY name is Eugene, aand i like math) some interesting idea came to my head today.
If dividing by zero returns an infinity, then zero is not an absence of number, but an infinitely small one. This lead me to idea that numbers can be arranged based on their size into a finite section ranged from zero to infinity, where 1 is the center. Such system describes number based on their size relative to 1. I wonder is there any math schools dedicated to something like this?
PS, See attachment for better graphical explanation, sorry for some inaccuracy, and thank you in advance)
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File Type: jpg IMG_3192_signed_m3.jpg (155.6 KB, 484 views)
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November 15th, 2013, 12:06 AM   #2
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Re: Zero-infinity number arrangement based on number size.

An interesting question. It's best not to think of infinity as a number and to accept that 0 is a number. If you play a football match and lose 3-0, then there is nothing mysterious about that score. 3-0 means your team didn't score any goals: it doesn't mean you have an absence of a score or your score is infinitesimally small. It means you didn't score any goals.

Early mathematicians, however, struggled with the concept of 0 - you can read all about that online.

Infinity is not a number (although there are ways to extend the number line to include points at infinity). In particular:



Instead 1/0 is "undefined". This means that you cannot carry out the operation of dividing by 0. If you look on this forum, you will seen things like:



This means that the function f is not defined at x = 1. You cannot write:

Careful mathematicians will always note when a denominator like this could be 0 and specify the values of x for which the equation is not valid.
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November 15th, 2013, 02:33 AM   #3
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Re: Zero-infinity number arrangement based on number size.

Quote:
Originally Posted by Pero
\f you play a football match and lose 3-0, then there is nothing mysterious about that score. 3-0 means your team didn't score any goals: it doesn't mean you have an absence of a score or your score is infinitesimally small. It means you didn't score any goals.
A little off-topic but I had a great argument with my college roommate once. A baseball announcer said there was "no score" for a nothing-nothing game. I said that this was wrong; and in fact that there WAS a score, and that the score was zero to zero. It was not correct to say that there was no score. If there was no score, then there would literally be NO SCORE. But that's false because there IS a score, namely zero to zero.

My roommate, who was going to law school, vehemently disagreed. To this day I don't understand his logic. He thought you went from "no score" to there being a score, when someone scored a run. But I felt, and still do, that there is always a score. Initially the score is zero to zero; and the score changes during the course of the game.

Well anyway OP, the modern point of view is that zero us just a number. It's a point on the number line. It's special in that it delineates the negative numbers on the left and the positive numbers on the right, with zero itself being neither negative nor positive. (But it is even, a frequent point of confusion on the Internet.)

So zero is just a real number like any other real number. It does have a very special property in being the identity element of the additive group of the reals under addition. But in the end it's just a regular old real number.

Infinity is not a real number and it cannot be combined with real numbers. But we can invent a system called the "extended real numbers" in which infinity is defined as a symbol that has certain properties and can be combined with regular old real numbers in certain ways. For example if x is a real number, then x + infinite = infinity.

However, infinity divided by anything is not defined. And division by zero is not defined.

We can deal with those undefined situations using the theory of limits; but there are many philosophical subtleties. We must accept and embrace infinitary methods, which takes us away from the real world and further into the world of pure mental abstraction. So that the penalty for using calculus is that we can no longer be certain that our theories apply to the real world. They do seem to be USEFUL, but that's a different thing.
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November 15th, 2013, 12:24 PM   #4
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Re: Zero-infinity number arrangement based on number size.

now i think it is useless to solve mystery of zero division without context, because zero and infinity can have different meanings.
though i suppose when we put equation like: x + inf. = inf. we mean that infinity includes all numbers, x already belongs to it and is unnecessary in equation, otherwise it should look like this x + inf. = x + inf.
In case with score board zeros are just points of origin, and they do not mean absence of score. For example you want to know "how better one team performed than another" and you divide ons score by another: 2-6, 6/2=3 "the second team was three times effective". But if we have a draw: 3-3, we divide 3/3=1, and thats okay - teams performed evenly. but if we have 0/0 it is nonsence from algebraic point of view, but still teams have performed evenly, and in that case we can look at the zero as infinitely small number, and in that case 0/0=1, and that makes sense.
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November 15th, 2013, 03:18 PM   #5
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Re: Zero-infinity number arrangement based on number size.

I find the initial post very interesting. Replies using "current" definitions are not appropriate in some respects. I must collect some thoughts and hope to post shortly.
(ok, I'm back)
Let's really get the party started...unfortunately the superscripting got wiped, realize e+, e-,infinity+,infinity- are all superscripted

I certainly would love to see concrete examples of problems introduced by the new "0" (in keeping with the following narrative). There well may be some, or many.
Stream of conscious thinking follows.
A few caveats. I'll use different terms to try to emphasize the changed thinking involved.
a) positive/negative let's say we use epsilon for the "new" 0. One should use e- and e+ if applicable to avoid sign confusion (and it is required by reciprocity definition)
b) people don't divide by 0 now mostly because of current definition. Yes, there are singularities in math systems (the tangent function comes to mind), see lower discussion.
Looking at limits,
1/1=1
1/.1=10
1/e+ = infinity+
makes consistent sense to me. Now, one must do a comprehensive impact study to see what else is affected. What has been traded to "resolve" the undefined zero division?
A lot depends on how one "treats" infinity. If any number in an expression "dominates" so to make other terms and numbers irrelevant, that's "dominant"
Similarly, the new "0" is "small dominant" ie "inconsequential" under add/subtract with the same thinking.

something + "inconsequential" = same something
anything * "small dominant" = small dominant (ie e, sign irrelevant)
something + "dominant" = dominant (ie infinity+ or infinity-)
something * "dominant" = dominant (ie infinity+ or infinity- )
"small dominant" + "small dominant" = small dominant (ie e, sign irrelevant)
"large dominant" + "large dominant" = large dominant (both items same-signed, so sign follows)
It gets a bit trickier with (infinity+ ) + ( infinity- ) as well as (infinity+ ) / (infinity- ) (1)
(infinity+ ) * (infinity- ) -intuitively would like this to = infinity- (2)
SO, we can now substitute in 1/e for infinity (as newly defined) and check for logical result
(1) becomes ALN* e- by definition is -1
(2) becomes ( ALN-) / e+ = -1
The last 2 requiring reciprocals, and every expressible number, no matter how large, has one. Defining e and ALN as reciprocals is a trick, as conceptually, if one knows ALN is 10^200, then accepting e as 1/(10^200) is an acceptable "inconsequential" value. If someone takes the high end out further to 10^20000, e is similarly still just (albeit different) inconsequential value.
So far so good, the only thing I see is the new "0" requires , in lieu of the old infinity (as an etheral non-number quantity), replacing it with ALN (absurdly large number) or similar. Specifically, meaning taken as large (in absolute terms) as the computation (or computer) is willing to go (or until a limit/asymptote dictates results).
I can think of no specific instance where eliminating infinity as described would pose an issue. But, there are so many fields of math that I am in no position to say.
What does the new system really do?
all division is now defined (no singularity)
all reciprocals exist
all powers of 0 now defined
this has required altered definition of the largest and smallest ends of the absolute number line, and elimination of ever using "0" as now defined. This sort of fits with the classic paradox about approaching a wall by halving ones distance to it recursively, and never reachng it. You do now "reach it" as newly defined, as you can always reach any e+.

How has calculus, series and other areas been affected?
Let's go back to the trig functions cot and csc.
Now, cot(0) is really cot(e+) or cot(e-). These are now defined (and match with the plot!, heading off to +/-ALN).
Even better is csc. Look at the plot, its symmetry is clear. Just as csc(pi) = infinity+ now, so will csc(e+).
Recall csc=1/sin and cot=cos/sin. sin(e+)= e+ so csc(e+)/sin(e+) = 1/e+ = infinity+
We must force ourselves, just as we never write csc(infinity), to now never write f(0), rather f(e+) or f(e-). One should also never lump ( infinity- and infinity+) simply as "infinity", as they are 2 distinct numeric areas. This plays a role in Calculus limits (L'Hopital's rule etc). Whereas, e+ and e- do approach the same location.
The "old" 0 is still located right where it is, but it would never be treated per se in math equations. It is still even, by its location midway between other evens etc. It is still a "number". Again, it relates to the paradox of measuring a coastline, whose length always changes as one zooms in finer and finer.
what of the old "0^0" ? every power of 0 returns 0 except for 0^0 is undefined [though checking in Pari/GP it returns 1, I presume icw (every other number)^0 = +/- 1 ]
Newly defined, we can have ( e+)^0 =1 like all other values...
0.01^0=1
0.0001^0=1
0.00000000...1^0 = ( e+)^0 =1 (akin to the classic 0.9999... = 1 logic)

SO, my very quick analysis has identified no immediately obvious inconsistencies.
(2) pertinent links:
http://en.wikipedia.org/wiki/Undefined_(mathematics)
http://www.askamathematician.com/201...ist-in-nature/
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November 15th, 2013, 09:50 PM   #6
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Re: Zero-infinity number arrangement based on number size.

Quote:
Originally Posted by Maschke
A little off-topic but I had a great argument with my college roommate once. A baseball announcer said there was "no score" for a nothing-nothing game. I said that this was wrong; and in fact that there WAS a score, and that the score was zero to zero. It was not correct to say that there was no score. If there was no score, then there would literally be NO SCORE. But that's false because there IS a score, namely zero to zero.
This is a matter of definition, of course. But any mathematician would agree that your choice is reasonable and your friends' is unreasonable. The idea of having a "right" definition is philosophically interesting but extremely common in mathematics.

Quote:
Originally Posted by Maschke
Infinity is not a real number and it cannot be combined with real numbers. But we can invent a system called the "extended real numbers" in which infinity is defined as a symbol that has certain properties and can be combined with regular old real numbers in certain ways. For example if x is a real number, then x + infinite = infinity.
Right. To be pedantic, though, you should write +infinity when you're talking about extended reals, since they have two nonreal elements +infinity and -infinity. Compare the projective reals which have just one, infinity, which is neither positive nor negative.

I've come to prefer to say that +infinity is not a complex number since it avoids the unfortunate connotations of "real" here.

Quote:
Originally Posted by Maschke
However, infinity divided by anything is not defined.
In the extended reals you're allowed to divide either infinite element by any nonzero finite number and get back the infinity of the appropriate sign.
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November 15th, 2013, 09:53 PM   #7
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Re: Zero-infinity number arrangement based on number size.

Quote:
Originally Posted by Omnispark
now i think it is useless to solve mystery of zero division without context, because zero and infinity can have different meanings.
It is certainly true that people use "infinity" to mean many different things, and it's not very useful to talk about it until you specify which you mean.
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November 15th, 2013, 09:55 PM   #8
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Re: Zero-infinity number arrangement based on number size.

Quote:
Originally Posted by billymac00
a) positive/negative let's say we use epsilon for the "new" 0. One should use e- and e+ if applicable to avoid sign confusion (and it is required by reciprocity definition)
b) people don't divide by 0 now mostly because of current definition. Yes, there are singularities in math systems (the tangent function comes to mind), see lower discussion.
Looking at limits,
1/1=1
1/.1=10
1/e+ = infinity+
makes consistent sense to me. Now, one must do a comprehensive impact study to see what else is affected.
Look up "IEEE arithmetic" to find some of the research literature. The standard terminology for e+ and e- are "signed zeros".
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November 16th, 2013, 06:30 PM   #9
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Re: Zero-infinity number arrangement based on number size.

Quote:
Originally Posted by CRGreathouse
Look up "IEEE arithmetic" to find some of the research literature. The standard terminology for e+ and e- are "signed zeros".
Yes, thanks, I was aware, just trying to have a compact way to use in narrative. I have a few more thoughts but poking around I did find a text by Moritz Pasch (Essays on the Foundations of Mathematics, p. 163) where he says "....Infinity is the reciprocal of zero. Zero is the reciprocal of infinity...." (with further explanation).
Anyway, one still needs "0" in its placeholder role in a number like 2106 . A few other areas to think about are Diophantine systems and operations like root and extrema finding by setting expressions=0. As an aside, the image I use in my mind on this "redefining" is a line which approaches a cliff at either end (0, +infinity) , with a similar line to the left similarly approaching (-infinity,0) on the left&right respectively.

(-infinity cliff) |-----------(negative numbers)-------------------|(zero cliff) |---------------(positive numbers)----------------------| (+infinity cliff)

Of course in our current system, one can move across/through 0 without issue (which introduces the singularity issues as a byproduct)

enough already ...
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November 16th, 2013, 08:38 PM   #10
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Re: Zero-infinity number arrangement based on number size.

I prefer the image of the Riemann sphere, personally.
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