Math Philosophy-- Why does 1/∞ not equal 0, and for that matter, what is ∞?

reply to post by Sly1one


Well, every point of space this man travels has an infinite number of points in it, so even the first 25 feet that he has walked, he has kind of already traveled an "infinite" distance of sorts.

Originally posted by PhysicsAdept
reply to post by Sly1one

 

Well, every point of space this man travels has an infinite number of points in it, so even the first 25 feet that he has walked, he has kind of already traveled an "infinite" distance of sorts.

Exactly...which is why all numbers are arbitrary in the presence of infinity.

reply to post by OutKast Searcher


Weeeeell, no I don't think that is quite right.the limit does not approach zero. By definition, the limit is zero. the graph of y=1/x is the one that approaches zero, but as x approaches infinity, the limit approaches zero which is where I think you may be confused. IF (if) x was to equal zero, then the limit would equal zero... well at least how I see it. But because of the limit itself, it doesn't approach zero. Approaching is already in the definition of a limit, so the limit itself does equal zero. I am sure you agree with this. I understand what you are saying, really I do.

But, here is where I disagree. Now I know you can't define any equation with infinity in it, but here is where I will try to make my case a little bit louder. What is 1/100? .01. What is 1/1000000? .000001. What about 1/1E25? 1E-24. You can keep doing this until you reach the biggest number. The biggest number, we will regard as being not infinity but some very large number. [Don't stop reading here, this is speculation just as much of what else has been said against what I am saying ] So you have the largest number, which is smaller than infinity. You keep dividing one by that number until you get the smallest number... the point where that number cannot actually get any smaller. This is where if you were to zoom in so far that actually zooming farther would not be possible. You divide one by just one more than the largest number... perhaps zero would pop out?

Eh ok that explanation is a bit skakey. But, as someone else in this thread stated previously, there is a proof to show that .9r=1. Bascially, you could say that 1-.9r equals 0, could you not? At this point, according to this basic mathematical proof (which I have also supplemented by using the summation model), we could say that .0r1 must then equal 0... and 1/∞ in theory would equal .0r1... or 0.

Take it or leave it, but I suppose there needs to be someone on each side of the fence for good debates to occur

At the very least, tell me you disagree--but admit that what I just said makes sense.

reply to post by Sly1one


Yep. And that is where math can no longer do the talking. It is too situational.

reply to post by OutKast Searcher


Ah yes, we may have erred there. I can agree with you there. Hexadecimal itself goes from 9 to A, then B, all the way up G right?

And about the pie, with the two or three people that have mentioned it. If you cut a pie into three parts, there is no law guaranteeing them to be equal. Graphically, maybe they can.... but it the real world even if you got it really close to being equal it would be more likely that you would cut it into 3 pieces of equal .33 of a pie, and then one piece of .34 of a pie... Just saying. Math is exact (usually), yet the world cannot be.

You might find this Horizon documentary interesting, 1 hour talking about the peculiarities of infinity.



It just made my mind boggle.

Originally posted by PhysicsAdept
reply to post by OutKast Searcher

 

Ah yes, we may have erred there. I can agree with you there. Hexadecimal itself goes from 9 to A, then B, all the way up G right?

Hex goes up to F.

In a Base n system, the digits go up to n-1.

Binary, base 2, the digits go up to 1

Octal, base 8, the digits go up to 7, after 7 comes "10" which is "8" in base ten.

Hex, base 16, the digits go up to F.


I'm not ignoring your other post...just don't have time to respond right yet.

Like you I like to play with numbers and I ran into one I call a thom unit which I mention only because it challenges the concept of an infinite number of points on a line which many are claiming/using to justify their views.
Follow:
A point has no dimensions, only coordinates.
Any two points with the same set of coordinates are the same point
A thom unit is the minimum distance two points must have between them to maintain their integrity as two separate and distinct points.
The problems:
1) Since we are discussing a length in a 3-dimensional world it must have a positive value. Two points can not be -3 inches apart.
Since it is the "minimum" distance two points must maintain to remain distinct it follows that it can not leave enough room to insert a point in between. Therefore it's length must be shorter than the length (width) of a point which is zero.
So now we have a positive number that is less than zero?
Exactly where should I try to locate this on a number line?
2) Would not the number of points in any line then be expressed as length/thom?
Just playing but I thought you might like it.

There are probably more than one way to tackle this problem in a good way. But this is how i try to teach others to get a visual perspective of what the infinite "is". All you need is a sheet of paper and a pen.

The sheet of paper represents the infinite. The infinite is the absolute first dimension. It is number 1.

If you draw a dot on the sheet of paper, the dot becomes the second dimension (number 2). In math we are not thought to think this way, or view a problem in this way.

We are thought that the first "dot" on the sheet of paper is the first dimension. But that is not correct in the terms of what dimension was first.

You have to agree that the sheet of paper is it's own dimension? And you would absolute need it to have the second dimension (the dot)?

You also have to agree that you can only draw the second dimension (dot) on the sheet of paper?

Before you start to draw or write on the sheet of paper. Ask your self:

-How can the sheet of paper form a dimension without you making it? How can the infinite form a new dimension on its own?

It is a very important question to ask, to get a perspective of what you are dealing with.

A clean sheet of paper has no dots (energy mass). So you have to visualize and think a bit back to the knowledge of what a vacuum is. A vacuum is a space (dimension with less matter than atmosphere). A absolute vacuum is a space with absolutely no matter (energy mass) compared to the atmosphere "Your very first dimension".

Math doesn't cover the very first dimension physically. Because we can't create the absolute vacuum and study it. So yo have to start to troubleshoot different ideas and solutions to get a new perspective to work with. And you can use math to control you troubleshooting. But math can only physically relate to dimension 2 and 3 and so on.

We have different types of energies. We have energy, energy mass and emitted energies. And than you have the infinite. All the different types of energies exists "within" the infinite dimension. Non of the energies can be the same as the infinite. That means the infinite can not be energy. It must be something different. Because the infinite is neutral. "Its a constant". Energy is not something that is constant.

I need a break o make cop of coffee . I will be back to explain more.

I have thought about this before, like while measuring a board to cut. You try to get exactly on the mark, but if you could use a magnifying glass you would find the mark is quite wide.

I think that trying to divide one by infinity is like trying to measure an atom with a yard stick. Infinity is not a real number, it is just a way of saying that compared to one the number we are looking for is really really small. When measuring in values of 10 to the negative forever I would think you would be also using comparative values. Because if a number is at the .0(infinitly repeating 0s)1, a value of one would seem like an infinitely large number. To an ant an inch is a mile. Numbers are just values we assign to measure our world.

So I would say that no, it doesn't equal zero, but compared to 1 it might as well be. So in a real world application, because the values are to far apart, and the result is close enough to zero to be zero.

reply to post by OutKast Searcher


No that's fine, thanks. I never really got into hexadecimal much, the letters frustrate me. I know about the other bases pretty well, do you have any idea why we are primarily a base 10 society anyway? Someone in this thread mentioned it was because we had ten fingers, which does seem likely I suppose

reply to post by PhysicsAdept


I'm no math expert but I believe that there is no way that 1 can equal 0. If for no other reason than the fact that 0 is not a number technically speaking. O o yes I said its not a number. From the math theory I have read 0 is a place holder only, nothing more nothing less. It helps us denote the difference in items like 1 item versus 10 items. Very handy indeed. But is it a number? According to those that do math theory, no.

reply to post by Puck 22


That is interesting, if it was true. Well interesting anyway I suppose. Yet I would like to remind you that a neagtive sign has nothing (usually) to do with something being small. It is a way to determine the direction of something. In physics, quite simply, it means you are going backwards or lost something opposed to gain it

reply to post by Dimithae


Well zero both is a number and is not as far as I see it. Like infinity, it is a concept IMHO. I say that because there is no such thing as nothing... except for in concept. There will always be something, even if it is small. The only way you could ever get to nothing is by, so I don't know, dividing by infinity??

reply to post by PhysicsAdept

I do not mean this offensively I assure you but your argument seems to be based upon semantics rather than math.
The problem can easily be rephrased without the use of the term 'negative' or 'negative number'. Simply replace it with 'less than zero"
That it represents the length of an actual distance it must have a value greater than zero but to maintain that it is the minimum distance two points can approach each other without becoming the same point then it follows that it must be 'less' then the width of a point which having no dimensions has a width of zero.
And so now we have a unit the length of which must be both 'more than zero' and 'less than zero'
Think about it, which was my only point. I do not wish to derail the thread but I saw so many replies that used an "infinite number of points in a line" I just thought I'd throw that in so just to play with some heads and suggest that maybe the number of points in a line is length/thom. Just a teaser for those who enjoy such things.

reply to post by PhysicsAdept

Weeeeell, no I don't think that is quite right.the limit does not approach zero. By definition, the limit is zero. the graph of y=1/x is the one that approaches zero, but as x approaches infinity, the limit approaches zero which is where I think you may be confused. IF (if) x was to equal zero, then the limit would equal zero... well at least how I see it. But because of the limit itself, it doesn't approach zero. Approaching is already in the definition of a limit, so the limit itself does equal zero. I am sure you agree with this. I understand what you are saying, really I do.

I said you should read the answer as the function 1/x approaches zero as x approaches infinity....the "answer" to the limit function is zero. But that does not mean that 1/infinity is zero. 1/x will never ever get to zero...it keeps approaching zero...but it never gets to zero.

When you say "IF (if) x was to equal zero, then the limit would equal zero... well at least how I see it."...I think you meant if x was equal to infinity. But IF x was equal to infinity...you would no longer need a limit.

But this is all a moot point because you can't treat infinity as a number....1/infinity is not defined just as 1/large is not defined.

What is 1/100? .01. What is 1/1000000? .000001. What about 1/1E25? 1E-24. You can keep doing this until you reach the biggest number. The biggest number, we will regard as being not infinity but some very large number. [Don't stop reading here, this is speculation just as much of what else has been said against what I am saying ] So you have the largest number, which is smaller than infinity. You keep dividing one by that number until you get the smallest number... the point where that number cannot actually get any smaller. This is where if you were to zoom in so far that actually zooming farther would not be possible. You divide one by just one more than the largest number... perhaps zero would pop out?

There is no such point to where you can not divide it further in math theory.

In reality, it depends on what you are talking about...and in reality things are finite, not infinite.

And you can not divide things down into segments of zero...because you can't add a bunch of segments of zero back up to a finite length. For example...you have a line with a length of 1....no matter how much you divide it, each segment will have some length. It may be very very small...but it will always have length.

But, as someone else in this thread stated previously, there is a proof to show that .9r=1.

Because we are working in base 10 and not everything works out nicely in base 10.

At the very least, tell me you disagree--but admit that what I just said makes sense.

I can't say it makes sense because it is breaking basic math concepts.


I am a bit dissapointed you didn't address any of the examples I gave. I would be interested in your thoughts on the following.

1) The limit of ( sin x ) / x as x approaches 0 is 1. But I know you would never claim that ( sin 0 ) / 0 equals 1...because you know you can't divide by zero.

2) If you think 1/infinity equals zero because of the limit function. Then you also believe that 10/infinity equals zero...because the limit of 10/infinit is also zero. This introduces a pradox for you.

1/infinity = 0

10/infinity = 0

1/infinit = 10/infinity

1 = 10

And it doesn't stop there...if what you want to say x/infinity = 0 for any number...then you are esentially saying that x = y for all real numbers where x and y are different numbers.

You know how they say a picture is worth a thousand words... well here's a few.



Or if you prefer decimal:

Note how the "root circle" acts as the termination points where numbers are multiplied with their reciprocals.


Showing polarity:

Note how ∞/∞ and 0*∞ are on a polarity border, making the result +/-1 (at the same time).


Conventional math is crap. Schools need to be updated with this stuff. And you aren't even seeing it in 3D here.

Infinity works just fine as a number, like 0 does. If 0 is considered "a number" then so should infinity. You can call it a limit if you really want to, but IMO that's semantics.

0/0 works fine too but it requires switching the origin from 0 to infinity (a simple change in perspective - click the 3rd image to see). Only defeatists say it's "undefined".


I didn't mean for this to turn into an epic rant, but I hate how even college professors with all their PHDs deny infinity's existence or can't figure something out so they say it's "undefined". It's a real sore spot with me. I taught myself math with a 3d modeling program and a calculator so I have a unique perspective on it. I didn't even have a proper elementary education (homeschooled without a teacher from 4th up), but that link in my sig didn't program itself!

As I began integrating my perspective with the conventional, I was rather disappointed in the current level of education in math (or lack thereof) in this world and in particular how credentialed people are misleading the masses with their ignorance while acting as authorities on the subject. As someone with very little education I expected all of this (and more) would be known already.

reply to post by circlemaker

Infinity works just fine as a number, like 0 does. If 0 is considered "a number" then so should infinity. You can call it a limit if you really want to, but IMO that's semantics.

If that is true, please tell me the answer to these two problems.

1/infinity = ?

2/infinity = ?

reply to post by OutKast Searcher


Anything / infinity = 0, except infinity. It's hits the limit. In this sense "limit" is used in it's proper context but I still say infinity's a number. I think any number can be treated as a limit. I've seen how 1, the square root of 2, the square root of 5, and phi are also limits (or mirrors as I like to call them). Oh yeah and there's 0.