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

Ok so first I am doing to divulge background information, of myself with everyone. Yeah, I do think it is important but if you do not care, skip this next paragraph.

I love math. I am in high school and I understand that I know nothing. I have taken basic Physics, Advanced Chemistry, Advanced Statistics, as well as Calculus II (my favorite subject ever). I sit at my desk for fun and contemplate math and everything that exists conceptually within it. Generally, these periods of fun generally end in me writing a 5-10 page essay of my thoughts. Unfortunately, threads that I have authored about math generally end up with people telling me I have no idea what I am talking about and that, basically, I am trying to be smarter than I really am. One, day, I hope I do know a great deal of math. I am going to go to Colorado School of Mines and become a Physicist. I WANT to know things. I think of things that many times scare me, involving math. I try to theorize and, well, I have no place to do so when I know nothing. Here is where ATS will help me, I hope

Ok, so I just wrote a thread not too long ago, in which I was trying to describe how I believe some infinities must, in concept, be larger than others. Within this thread came up the dilemma of 1/∞, and how I think it must equal 0.

Now I am versed very well in calculus and understand the concept of limits very well. A limit merely tries to show in a graph what appears to happen, even though it may not actually occur. So if you find a limit of something, you are using math to show what is going to happen at a point where there is a hole in the graph, an asymptote, or maybe even how you think the graph will behave after a long (a very long or even infinite) period of time.

A good introduction to limits
A deeper understanding of limits, and how to evaluate them

Now, the question is: what happens to 1 when it is divided by infinity? Well you can look at it graphically in the example of y=1/x, and take a look at the limit of y when x approaches ∞. Well, you end up realizing that the limit is zero. So does this mean that 1/∞ equals 0? Well according to many people, the answer is no. 1/∞ is undefined and actually equals a number that is (relatively equivalent to) 0.0(repeating)1. Well this makes sense, and you cannot dispute this, really in any way that I know of.

But I say, what if you pretend that an infinite time has passed for y, and that we pretend that x actually does equal infinity? Then, wouldn't be look at the limit, and assume that because the limit as x approaches infinity equals zero that if we reached infinity, it would be equal to zero?

Now, we cannot treat infinity as a number--I get that. It has no numerical value and is really more of a concept than anything. It cannot be treated as a number, but can still be examined, IMO as one, in math (you do this informally with L'Hospital's Rule all the time).

Here is a good explanation of infinity, through diagrams

So I have tried to put together some basic information of infinity, but I have also come across some very complex explanations of the "plane of infinity". I most want to explore this aspect of math, because I think it will help me grasp what I am trying to communicate with the difference of infinities, and the imaginative prospect of actually looking at something as it exists in an infinite state...

So, ATS, wanna help? I need some people to go through with me and others here to decipher this math and help clear things up once and for all. Here are the links to what I am talking about, clear explanation on any of them are useful:

Projective Infinity
Transfinite Numbers and Set Theory
Line at Infinity
Point at Infinity


My biggest priority here is to understand. I hope others can respect my mathematical ignorance, I have a desire to know though.

Also, here are my failed attempts at trying to know something if anyone is interested in furthering their mathematical thoughts:
Limits at Infinity, the number 1 is bigger than you may think
Clarification on Time and What Creates it (*Note, this one was written within my first week or so of Calculus so the math part isn't the most beautiful of things.

So the first thing I would like to point out, is Euler's number, e. e=2.72828... and represents lim x->∞ of (1+1/x)^x. Now the "fundamental" flaw of what I believe people are thinking that I am not grasping is that if I plug ∞ into this equation, assuming that 1/∞ equals 0, then you get (1+0)^∞, which would logically equal 1. So I see that flaw there, but I still need more proof. If 1/∞ is undefined, that mean it can equal more than one thing at any given time, right?

Since you are interested in infinity, check this thread.

www.abovetopsecret.com...

o WAIT.....
Remember, math is a set of rules we use to describe the world. At times, it breaks or doesn't make sense(negative distance). You'd do well to focus on the mechanics and applications of math. Analyzing the strength or hardness of a hammer is useless if you never pound a nail.

Edit: Love golden sun btw. Nice avatar.

This post is like the joke about the 50 foot giant, WAY over my head.

The concept I have of zero is nothing, or on a number line the point of origin. As for infinity, what I grasp is that it just keeps going, like the energizer bunny.

I may be wrong, I am a high school drop out (but still went to college ).

I find that the limit of math I really use is adding and subtracting from my checking account. Unfortunately I do almost the same of both.

If one divided by infinity is zero, then zero times infinity should equal one.

2nd

Originally posted by OccamAssassin
If one divided by infinity is zero, then zero times infinity should equal one.

2nd

Except one divided by infinity isn't zero.

The limit as x approaches infinity of 1/x is zero.

reply to post by MeesterB


Exactly!

*insert applause*

To add
Does 0.9r = 1 ?

reply to post by OccamAssassin


Right, and actually both equations I believe are considered indeterminate, where basically you cannot really know what each of those will equal until the correct situation presents itself... So if 1/ ∞ does equal zero, then in that same case 0*∞would equal 1...

reply to post by MeesterB


So then at infinity, would it not equal 0?

I mean, at a very large number it would equal close to zero but we are not talking about a number at all. We are talking about a concept. So then conceptually, would not 1/∞=0?

I don't think this is something that can be over-thought, but I do think that there is some answer out there that will make it more clear...

including infinity in equations and logic causes singularities/paradoxes to arise.

if infinity is truly infinite, than zero times infinity is the mathematical equivalent of "what happens when an unstoppable force meets an unmovable object?" the answer is that both cannot exist at the same time.

reply to post by PhysicsAdept

Right, and actually both equations I believe are considered indeterminate, where basically you cannot really know what each of those will equal until the correct situation presents itself... So if 1/ ∞ does equal zero, then in that same case 0*∞would equal 1...

Ahh ....but if I have 0*∞ apples(those damn apples again) I don't end up with one apple.

Originally posted by OccamAssassin
reply to post by MeesterB

 

Exactly!

*insert applause*

To add
Does 0.9r = 1 ?

0.9r=1
=> 9/10r=10/10
=>r=10/9
I don't see what you are getting at.

reply to post by Bob Sholtz


Right, that makes a bit of sense. Because then you end up getting the possibility of that equation ending up to be 1/0=∞... So are you saying that infinity cannot exist where zero exists?? Basically, 1/∞ is really an identity of it self, and is equal to only 1/∞?

reply to post by OccamAssassin


Then what do you end up with? Because it is an equation that cannot mathematically exist as the same answer each time.

Originally posted by PhysicsAdept
reply to post by MeesterB

 

So then at infinity, would it not equal 0?

I mean, at a very large number it would equal close to zero but we are not talking about a number at all. We are talking about a concept. So then conceptually, would not 1/∞=0?

I don't think this is something that can be over-thought, but I do think that there is some answer out there that will make it more clear...

Yes, but you can never reach infinity so you have to use a thing called a limit.
infinity isn't a set value so you can't just plug it into an equation like you are trying to do.

reply to post by PhysicsAdept


Here is the way I look at it and why it doesn't equal 0. View yourself as the 1 and the universe as the infinite. You don't equal zero just as the 1 wouldn't equal 0. Also, everything, and I mean everything in this universe we are in is infinite so to your second part yes one infinite can be larger than the other its just how you look at infinite. Infinite can go both ways either infinite in contraction or infinite in expansion. Some may or not agree with me but that is how I feel it works. so you yourself are infinite just as a piece of dirt next to you is infinite and the atoms that make it up are infinite and so on and so on.

reply to post by PhysicsAdept

I don't think this is something that can be over-thought, but I do think that there is some answer out there that will make it more clear...

But ,but, but.....that would take all the fun out of maths.

BTW S&F for another cool thread.

Thank Christ, my wife has never put the hard word on me......Maths or her......I don't think that she would like the answer.

reply to post by MeesterB


Exactly, I get that. When I talked about Euler's formula I pointed that out, but I am asking what would happen if we did. Not to use it as a number, to but pretend that an infinite amount of diving had occurred... Open your mind to that possibility... then, what is the answer? Would you not then look at the limit? The limit is saying, indirectly, that at x=∞, 1/x=0

reply to post by PhysicsAdept


Personally, I don't think you understand the concept of infinity.

Now, the question is: what happens to 1 when it is divided by infinity? Well you can look at it graphically in the example of y=1/x, and take a look at the limit of y when x approaches ∞. Well, you end up realizing that the limit is zero.

how can anything "approach infinity"??? how do you know you are there? You can't...there is no approaching anything in infinity...its like traversing a circle, you will never get to the "end" and it doesn't matter where you start as start/end are man made points of reference to quantify something for measurement. Start/end in the objective sense doesn't exist...unless you the person decide to assign points to something and that is all your imagination at work.

1 is a man made concept/digit...it doesn't truly exist anywhere. Go find me "1" of something...once you think you have I'll show you more, or divide it and call it another digit....

give me 1 apple ill show you an apple tree, give me one rock ill show you a beach, give me 1 sun, ill show you a galaxy, give me 1 galaxy ill show you a galaxy cluster, give me a galaxy cluster and ill show you the universe...

1=∞
in which case your equation would read as such:
∞/∞=∞

reply to post by MeesterB


Sorry I should not have forgone the character map.

r = reoccurring

0.999999999999......r

Think 1/3 = 0.333333333r

3 times 0.333333r = 0.999999999r = 1 = 3 times a third.