Math Is Fun Forum

I don't mind shaking things up a bit here.

I don't think that the basic concept of infinity require a masters degee to understand.

And I am certain that there is honest debate and not half the established consensus on some of the things I put forth as you make out to be.

Some things are not 'surface intuitive' and require an imagination. Some things I think are flatly intuitive. Grasping infinity can be incredibly difficult, and is only made easy when you accept it's not a number, but then it can become very easy. But I do know that the idea of infinity is to express unlimited and all-encompassing, so when I see it in relation to numbers, to me it means one simple thing (an impossible end to numbers and the concept of labeling all potential quantity with a name). It can be thought of as a package outside of which can exist nothing, and inside of which exists everything that can ever exist.

The fractions I put forth are nonsensical. They don't pretend to be numbers, and I'm not meaning them to be or claiming them to be numbers. I'm sorry I gave that impression. A number is a finite, measurable quantity (adding further...located between the infinitesimal and the infinite both of which are not numbers either).

Any fraction that has infinity involved with it, by definition cannot be a number. I have no disagreement and I believe I understand why that is for the same reason you do.

I'm trying to adapt a concept to a shorthand way of expressing it because some things seem unnecessarily muddled, possibly by overthinking them.

I do believe I have a very good understanding of infinity, which is not a number.

I disagree that 1/infinity is not infinitesimal...there are probably as many people defending me on that who are much further along in math than I am, than people who are defending the idea it is not. And why doesn't that make sense to you? Even the number 1 when used with infinity is symbolic and not so much a number at that point, because any finite number will do when placed next to infinity (they all appear to converge on zero). Perhaps we can focus on that piece first, since it's a very interesting subject.

I don't believe zero is a number...but I have a slightly different idea of a number in that case.

I think of a number as a existent quantity, so as to distinguish it from a 'quantity of zero'.

OK, you are not counting to a finite quantity. You are counting to no quantity. Show me zero dollars and zero cents and then explain why I should accept you have SOME (a finite quantity) of money.

Well, you can't because if you did, it would be at least a penny in this case. Zero dollars is equal to zero apples is equal to zero planet Jupiters. If those things are absolute equals (they are) then zero is not a true number.

Zero has a unique characteristic that sets it apart from numbers we normally think of as numbers. One or two apples, ten apples, half an apple, we're still talking apples...zero of something...what are we talking about? Battleships? Could be. Does it really matter at that point? No. It matters only if there's a number of something involved, meaning a non-zero quantity.

In order for any of those three things I mentioned NOT to be the same, they have to be a finite quantity, something other than zero. Zero does strange things when you insist it's a number. Why not have infinity be called a number? You can put it on a number line also. I don't think that's the sole justification for zero to be a number.

Like I said, it does have a place on the number line...I'll never disagree why it's there...I just disagee on what it is. It's a very underappreciated entity in mathematics.

Its basic identity is 'diminished' when people insist the infinitesimal is the same.

True, infinity is not a number...neither is zero. That would be my argument. In order to be a number, it must exist and also be in the finite realm.

Zero has a problem...it doesn't exist...Infinity has a problem...it's not finite.

These ideas work together well in pairs and are opposites of each other:

Zero and infinity -- no quantity whatsoever and all quantity

Infinitesimal and infinite - approaching zero and approaching infinity

I roll my eyes everytime I see someone saying ".000 ... 1" equals zero. It makes as little sense as someone saying "999 ..." equals infinity.

The idea of marking the difference between being the extreme and approaching the extreme by allowing those identities are nothing to be ignored,
and I think you cannot understand zero and infinity correctly without accepting them.

Question:

Does the mathematics community look at the number line the way I do, thus (I'm only using the positive half, but it also applies to the negative half):

5 basic representations (for lack of a better word) of numbers in a traditional number line (aside from positive/negative):

1. zero (itself)
2. infinitesimal (forever approaching, but never reaching zero)
3. all finite numbers between the infinitesimal and the infinite (including all fractional, irrational and trancendental numbers)
4. infinite (forever approaching, but never reaching infinity)
5. infinity (itself)

I have seen some discussion where people insist that zero and infinitesimal, and infinity and infinite are virtually
the same and allow approximations to zero and infinity, but I think that leads to errors in understanding them.

Literal zero and literal infinity should be allowed to stand alone without approximations. They both represent a pure idea.

I understand there is a proposed symbol for the infinitesimal (epsilon - ε). I like it.

Is there also a proposed symbol for the infinite (which, as far as I know is best represented by ∞ - ε, not ∞ - 1)?

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Last, this part is interesting about the number line (I wrote this last night):

(Let me know if you think this is off base, or I'm not quite
describing the idea I'm trying to convey in the best way.)

Say you have a number line as such (note that there are no arrows on the ends of the line, the ends of the line are in fact - and + infinity):

-∞ ---------------------------------------------------------------------------------------------------- +∞

Q: where would you put the zero?

A: anywhere you want to in between; infinity has no scale
   so you don't have to worry about exactly centering it;
   every possible number in between would crowd adjacent
   to the zero anyway, making it impossible to plot any
   other number, no matter how large it was; the other
   numbers would neither be on top of the zero (equal to
   it) or away from the zero (visually indicating what
   portion of infinity they are)

Q: after putting in the zero, can you put in any other numbers?

A: no; it is impossible to put any other number on the
   line, if you want to avoid imposing a scale; if scale
   is not important, you can do anything you want;
   the reason why you cannot put any other number on the
   line, is that you cannot get close enough to the zero
   to avoid the problem of dedicating a portion of the
   line to that number that visually indicates its
   proportion of infinity - that would not represent the
   nature of infinity accurately; and of course, you cannot
   put any number in the same spot as the zero, because
   you are saying that zero is equal to that number,
   which would not represent the nature of zero accurately

Q: instead of zero, can I put in a different finite number?

A: yes, but that is the only other number you can put
   on the line, (again, without imposing a scale)