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

reply to post by Sly1one


or you could say that because 1=∞, then ∞/∞=1. Then, because ∞+∞ equals ∞, then would not (∞+∞)/∞=1? No, because going back to 1=∞, you are then saying that 1=2, or basically ∞=∞.

Originally posted by PhysicsAdept
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

okay, lets just assume that you did.... then you would be wrong.
Math is beautiful because there are rules and a set answer. You can't just pick and choose rules and then point to the hole you just made as something profound.

reply to post by OccamAssassin


Ah I see now, does .9 repeating=1? Yeah I wonder that too... I read an article once and for the life of me I cannot find it... I figured it would make what I say to be more credible, so if anyone can find it they should post a link. I read once that because of limits, .9r actually does equal 1, at least in all practical senses. Does this mean mathematically we can call it 1? Probably not... because then 1 would equal 1.0r1, then that would end up reaching 2 at some point, basically stating that 1=∞, or ∞=∞ like someone else in this post has said...

reply to post by Sly1one


Personally I don't think anyone understands the concept of infinity. Try putting yourself into a position. This position is one in which you have just broken the infinite plane. That you exist somewhere, that is no ∞ away from where you started, and perhaps another ∞ past that point. Where then do you exist? This is where the philosophy has to come in, not necessarily the math.

But you claim to understand it well, or at least much more than I do, so then tell me what you think.

reply to post by MeesterB


Then refute what I am saying mathematically. The purpose of my original thread was based off the idea that at one point, you would be able to reach infinity. You can claim that I cannot reach it, but mathematically how do you prove it? Now I have quite a bit of common sense, and the definition of ∞ alone makes me think you can't just find its end, but that is why I posted on it. WHY?

And if you cannot prove why, then envision yourself at the end of ∞. Where are you now?

These are the questions of math that take speculation to answer, and I am curious in either a mathematical proof or the speculation one way or another that makes you and I agree.

reply to post by PhysicsAdept


I can see that your on your way to becoming quite a good mathematician. Keep it up.

In your first year at uni you will probably cover the 0.9r = 1 proof.

It is basically a way to show the value of using fractions in your work until the final answer is reached.

This will provide you with countless hours of joy as it minimises errors from rounding.

You will find that irrational numbers will get treated the same.....My prof's in uni preferred to have an answer like 2*pi/7 as opposed to calculating the answer in a decimal.

The proof is simple (I showed it in my last post)

1 = 3/3
1/3 = .3r
3 * .3r = .9r
therefore
1 = 0.9r

Something else to add.

You may want to look up a theory called the De-Sitter Horizon.

It involves calculating the speed at which the edge of the universe (is possibly) expanding.

It delves deeply into the infinity concept.

I think that you would find it particularly interesting.

De Sitter Horizon

reply to post by OccamAssassin


Yeah and that totally makes sense, and it pretty much clear-cut as far as I am concerned. It makes you wonder though. In a way, not mathematically provable, 1/3 is a form of infinity. It is a summation of 3/(10^n), from n=1 to n=∞. I don't know, I am sure you know that but it is something that is interesting to examine... ∞ in such a small way

reply to post by PhysicsAdept


The thing about this calculation is that in order to produce that result of 1/3... we assume we have reached infinity, have we not?

reply to post by PhysicsAdept

including zero in equations doesn't cause problems, but including infinity does. zero has a value, but infinity can't be accurately defined.

i don't believe infinity can exist within time. if time is removed, infinity naturally exists.

Originally posted by PhysicsAdept
reply to post by PhysicsAdept

 

The thing about this calculation is that in order to produce that result of 1/3... we assume we have reached infinity, have we not?

It appears to be infinite.....in decimal, base 10 form. Or as an infinite line of threes

What happens to our third as a decimal in base 3?

reply to post by Bob Sholtz


Well that is certainly an interesting aspect. Who is to say that zero is defined though? Zero has a type of numerical value that shows us that it is possible to have the opposite of ∞. I don't think that zero can exist without infinity honestly...

Originally posted by PhysicsAdept
reply to post by Bob Sholtz

 

Well that is certainly an interesting aspect. Who is to say that zero is defined though? Zero has a type of numerical value that shows us that it is possible to have the opposite of ∞. I don't think that zero can exist without infinity honestly...

Zero can be tricky to quantify.....In some contexts, it is a concept and in others a number.

Infinity is always(and can only be) a concept though......Just to throw a spanner in the works.

reply to post by OccamAssassin


Uhm, I don't know exactly what you are getting at, could you explain? I mean 3 in base three is merely 10, correct? I don't know how decimals work in other bases, but would it be.10r?

ETA: or would it be 1/3= 1/10, equals .1?...

Originally posted by PhysicsAdept
reply to post by OccamAssassin

 

Uhm, I don't know exactly what you are getting at, could you explain? I mean 3 in base three is merely 10, correct? I don't know how decimals work in other bases, but would it be.10r?

Spot on.

In base 3 it is 0.1.

Base 3 is odd and you will probably never use it beyond learning the concept in uni. I used it only because of the example.

is possible to have the opposite of ∞.

I was going to say -∞ but that is wrong as we can't have -0.

Thinking about it.......you are probably right....but......the opposite of 0 is the universe(including the creator if he/she/it exists).

Interesting concept......I think I'll be contemplating infinity for the rest of today.....

reply to post by PhysicsAdept

i wouldn't say that zero and infinity are exactly opposites. any number with a value is the opposite of infinity.
if we subtract 1 from infinity, we're still left with infinity. there is an infinite amount of numbers between 2 and infinity so that infinite's value can never be compromised by removing numbers. logically x-x=0, but infinity-infinity DOESN'T equal zero. infinity never ends or begins, so there is no way to subtract out of it.

if time is removed, then measuring in numerical segments becomes meaningless because there would be no difference between one foot, and one mile. that's why i think infinity can only exist as a concept in a place where time exists.

i've read theories on black holes forming event horizons because time is removed, and infinity is reached inside, so the event horizon is a kind of barrier that separates infinity from interacting with the rest of the universe via a point of no return.

Well...

1 / infinity is the inverse of infinity / 1; since neither = zero, zero is not equal to 1 / infinity or infinity / 1 or infinity itself.

Another thought... 1 = infinity, so any multiple would also equal infinity or any combination or fractional of any number except zero. Since zero is null and has no value, it cannot equal any combination or fractional of any "real number" except as a "place-holder."

But equally, zero can = zero.

reply to post by OccamAssassin


Here's a question I have wondered for a while... What is the purpose of other bases? I mean I know that we use decimal and it works out nicely because I am used to it, then base 2 is used for computers... probably because analogue things works off of a system similar to binary I am guessing, then there is hexadecimal. I don't know exactly the these bases are used instead of the others, but I am really curious to how the other bases benefit the world, what are the other bases even used for?

reply to post by Bob Sholtz


So because ∞-∞ does not =0, you would agree that some ∞s are larger than others?

ETA: also, to be noted, there are an infinite amount of numbers between 2 and ∞, but also 2 and 3.

reply to post by PhysicsAdept


Thank you for the invitation to join you in this thread. Now, I have to point out that I have virtually no knowledge of Mathematics. I left school with no qualifications per se, and that was that

I struggle with long division along with most other forms of mathematics principles, and just the thought of trying to get my head around Calculus is impossible. I truly do envy you, that you are able to understand and grasp the formula.

One thing I do seem to have is the ability to think logically, and the potential to 'grasp' understandings of things which most people never bother to try to rationalise. Kind of a philosophical thinker if you will.

Perhaps if I had ever been shown how to understand mathematics, I would be one of those thinkers that likes to dabble with calculations to try to work 'everything' out

So to get back to your thread, (and I will happily admit that I'm way out of my depth when it comes to the meat and potatoes)

Fairly recently, I watched this video and it intrigued me greatly. Specifically what Georg Cantor hypothesised with his Set Theory



At first, I took this all at face value but slowly, over time with my pondering over Infinities. I began to realise that in order for Cantor to be right. It would in itself prove that we actually live in an Infinite Universe.

Whether we live in a Finite or Infinite Universe is a different discussion, and a very complicated one at that. But fundamentally, Infinity itself cannot exist within a Finite Universe. Only as a concept.

Then it occurred to me that Cantor was putting boundaries around his infinities. He declared that if you draw a circle and divide it an infinite numbers of times, then take the lines from the sides of each angle and radiate them outwards past the circle edges to a larger circle drawn around the inner circle - there would be gaps because of the divergence of the lines.

This is where I state that Georg Cantor was wrong. Controversial I know, and who the hell am I ? an uneducated minion to declare a great Mathematician to be wrong ?

Well, what Cantor seems to have failed to realise is that despite the divergence of these lines, there would be no gaps! because new lines will 'Endlessly' propagate to fill the gaps, albeit at a slower rate perhaps than the smaller circle.

The gaps you're seeing are only there if you take an instantaneous snapshot of a single moment in time. Fundamentally though there can be no gaps unless you stop the propagation of infinity at which point it becomes finite. Kind of a paradox

But what we can 'possibly' conclude is that with 'Time' Infinity can propagate at different speeds ? OR does Infinity propagate instantaneously ? another question posed I guess.

Anyhow, to get back to your initial question of why 1/Infinity doesn't equal Zero

Someone asked this question

If I divide one chocolate bar between an infinite amount of people, everyone gets nothing. So where does the chocolate bar go ?

Here is the answer

The very sentence "1/infinity = 0" has no meaning. Why? Because "infinity" is a concept, NOT a number. It is a concept that means "limitlessness." As such, it cannot be used with any mathematical operators.

The symbols of +, -, x, and / are arithmetic operators, and we can only use them for numbers.

To write 1/infinity and mean "1 divided by infinity" doesn't make any sense. 1 cannot be divided by a concept. It can only be divided by a number.

In math, when you hear people say things like "1 over infinity is zero" what they are usually referring to is something called a limit. They are just using a kind of shorthand, however.

They do NOT mean that 1 can actually be divided by infinity. Instead, they mean that, if you divide 1 by successively higher numbers, the result becomes closer and closer to 0. If I divide 1 by a very large number, like a billion, then I get one-billionth, which is a VERY small number, but it isn't 0.

Since there is no largest number, I can always divide 1 by a bigger number. But that will just produce an even smaller number, right? It will NEVER produce 0, no matter how high I go.

Since the answer to the division is getting closer to and closer to 0, we say that "the limit of the expression is Zero." But we have still not divided anything by infinity, since that isn't a number.

You are perfectly correct in saying that "1/infinity = infinitesimally small."

But only if you realize that you REALLY mean "1 divided by a REALLY big number is a REALLY small number."

Sorry for some of the ramblings and I hope I have contributed something further to think about