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

reply to post by PhysicsAdept

Math is a 100% accurate representation of the universe like notepad is a 100% accurate image viewer.

reply to post by OutKast Searcher

I will have to see some examples of dividing by zero happening all the time, I still stand by that it can't happen.

Well, you know how Euclid's number e works, correct? Well really there are two different ways in which it is calculated, both methods produce a result and concept I would like to share.

1st, assume that f(x)=(1 +1/x)^x

"plugging in" infinity here results in 1^∞, correct? It is indeterminate so we use L'Hopital's Rule

take the ln of both sides and divide by 1/x to get an indeterminate form of 0/0

Use L'Hopital's Rule again and you get that lim x->∞ ln(f(x))=1. So, f(x)=e as x approaches ∞.

So yeah, here are limits again to decribe something's value... but all of math is based off of this limit process!

Here is another example that I have alluded to I think earlier. e^x= the summation from n=0 to n=∞ of (x^n)/(n!), another limit example... example of how we consider e to be a real number in math, yet it is just a limit with infinite decimals... So why then can't we use infinity? As a concept mind you, not a number, but a concept that illustrates a picture and a meaning? 1/∞=0 !!


I don't know, my posts are incohesive. When I explain things in person they always come out better. Answer this time, please, is any of this making sense to you? We may just have to agree to disagree. Your proofs aren't proving anything, and although I think mine are pretty good, you are stubborn enough not to move your position


It is not that I don't get the concept of infinity, it is that you say infinity cannot be reached. I say it can. I do not know how, or when but it must exist somewhere at sometime, perhaps at all time. BUT, that must mean we can treat it as a plane, no? Try proving to me that 0 exists--it too is a concept that many people mistreat and misunderstand.

Originally posted by OutKast Searcher

Originally posted by PhysicsAdept
reply to post by OutKast Searcher

 

Has to do with rates is my guess. You cannot just multiply each side by infinity to resolve that 1=2.

Ok seriously, last post of the night, more tomorrow

Why not?

So now not only is infinity a number, but it is a number that has different values at different times? And it doesn't follow basic algebra rules?

Well this may be a horrible example, but remember, we are dealing with indeterminates. So let us say that 1/0=∞. It is a vertical line and has a slope, also of some undefined amount. But then, rearrange the equation and you get 1/∞=0. Unless you would like to dispute this by saying 1/0 equals 1(0r). That is among the same principle you are using...

Originally posted by jb1958
Hmm, I think I can grasp infinity divided or multiplied by a number, but the OPs question, of 1 divided by infinity, I am struggling with. 1 divided by anything more than zero gives less than 1 right? a fractional number. And 1 divided by infinity? Seems it would be more than zero but less than one of something.

You hit the nail on the head with your last statement.

It would be more than zero but less than 1 of something.

i.e.

1/1 = 1
1/2 = .5

Yet there are inifinate numbers between 1/1 and 1/2. For example:

1/1.1 = .9090909090909091

-saige-

reply to post by saige45


Maybe a good way to visualize this, is we would have an infinite number of parts of something, but since there are an infinite number of them, you could never collect all the parts, so you could never make a whole something out of these parts. So no matter how hard you tried, forever, you have more than zero of it, but can't ever make a whole 1.

reply to post by OutKast Searcher

So in your world....100 = 1 and 45 = 0 and 32 = 98???? I'm sorry...but I have to say you are more confused than anyone else in this thread. I would suggest that you take some formal courses instead of just teaching yourself...easy to confuse yourself to the point where valid mathematics will never make sense to the incorrect world view you have built in your mind.

I don't think he is confused at all, I think that must be you. Or perhaps a little bit of everyone, myself included. You keep saying ∞ is not a number so don't treat it like one. So you cannot conclude 1=2 by this logic. We are saying that you cannot treat ∞ as a number either, but yet you can still use it in some numerical sense like 1/∞=0. Now here is a question for you... You say that 1/∞=.0r1, right? Then obviously 2/∞=.0r2? Or do you keep true to your sense of ∞ here and say that 2/∞=.0r1 in which case, you are saying 2=1.

Hmm.

You will never get an answer to the formula because ∞ is not a mathematical constant, it is a pure concept, an idea.

0.9r ≠ 1 because 0.9r has no limit; however, 0.9r≈1 if you set a limit and arbitrarily decide to round up 0.9r to 1

For the same reason 1/∞ ≠0 because ∞ has no limit; however, 1/∞≈0

Using the approximation of ambiguous or limitless formulas is acceptable in formulas where the consequences would not dramatically alter the results such as balancing equations or substituting equivalences. For instance, the value of π can never be calculated in its entirety but it is used, approximated, all the time.

Originally posted by jb1958
reply to post by saige45

 

Maybe a good way to visualize this, is we would have an infinite number of parts of something, but since there are an infinite number of them, you could never collect all the parts, so you could never make a whole something out of these parts. So no matter how hard you tried, forever, you have more than zero of it, but can't ever make a whole 1.

Yes and no.

As shown, 1/infinity has to include 1/1 as part of it's set. This means that it can be represented as 1 whole 1. However, it can never be 0. 1 can not be divided by something to make it 0.

-saige-

reply to post by Kemal

lim 1/x = 0 x->∞ limes means that the function gets closer to... (in the case of 1/x) to 0, but does not mean it IS 0.

Well perhaps, but in this case (does not work for ALL cases), would you agree that at infinity--if we could reach it, the limit would indicate of its true value? At ∞, would 1/x not equal 0? It isn't like an asymptote will just jump out of nowhere.

reply to post by ClydeFrog42

This has got to be one of the dumbest threads i have ever read.

This may be one of the dumbest, yet I think this one takes the cake: www.abovetopsecret.com...

I think you need to broaden your mind. When I said ∞ is a number alludes me, but I do know that math and the world for that matter is based off of concept. The whole numerical system is a concept, so deal with it. It isn't like I asked what 1/pizza was I asked a legitimate question. But if you really enjoy hating dumb threads, why even bother to reply? You obviously have better things to do.

In mathematics, if you have a function f(x) and take a limit as x approaches a value c, then the limit does not have to have the value f(c). Functions of this kind are special in mathematics.

You'll find that it was the great mathematician Augustin-Louis Cauchy who finally put limits on a solid theoretical footing. Georg Cantor figured out what infinity is all about.

Some things you will eventually discover:

1) Infinity is not a real number and so 1 / infinity is a meaningless expression

2) lim [x -> infty] 1/x exists and is 0

3) A function f(x) for which lim [x -> c] f(x) = f(c) is given a special name in mathematics. It is said to be continuous at c. Obviously, not all functions are continous, e.g. f(x) = 1/x is not continuous at x = 0

4) There are many different infinities, which are called infinite cardinals. For example, you can prove that there are more points on the real number line than there are counting numbers 0, 1, 2, etc., i.e. you cannot count all the points on the real number line. So the "number" of counting numbers is a different kind of infinity to the "number" of points on the real number line!!

So what is infinity? Well, it depends on the context. In the case of limits above, it just means "grows without a limit".

Once you accept that it is not a number (it doesn't obey any of the rules for ordinary arithmetic), the mystery about it goes away in large part.

reply to post by jb1958

If zero means none, nothing, nil, then you have none of whatever. zero apples times 3 is still zero apples, I still don't have any. But can infinity be divided or multiplied by zero, what does that mean? An infinite number of apples divided by zero, the zero becomes irrelevant relative to infinity's size, so the answer is still infinite? This is intriguing, what do other members think?

Well some would argue that infinity divided by anything other than infinity and zero would remain infinity. As for dividing infinity by zero... that is some answer that is too indeterminable.

reply to post by saige45


You did agree it would be more than zero but less than one, so maybe bad example I gave. So lets leave it at being less than one. My brain hurts from too much math anyway.

Originally posted by circlemaker

Originally posted by OutKast Searcher

I request a thread be made on "shadow numbers".

Maybe I'll present my counter argument of "invisible numbers".

No need for an entire thread. Count from infinity instead of 0 and you're dealing with shadow numbers. From 0's perspective they're all collapsed to infinity. From infinity's perspective all the numbers which use 0 as their origin are collapsed to 0.

Instead of offering non-answers I actually figure stuff out.

Here's a picture showing shadow numbers next to the "real numbers". The awareness barrier defines the numbers we haven't counted to yet, hence the name.

What circlemaker means by 'shadow numbers' is simply 'reciprocal numbers'; a totally new and untapped world of wonders and excitements, that no reader here will learn about in any school, college or university.

No readers, i reckon, would know f.ex. that the reciprocal number to the fraction 1/9801 contain all the numbers from 1 to 100 with the exception of the number 98 as in:
1 divided by 9801 = 0.000102030405060708091011121314....92939495969799000....1020304050607.... and the decimal expansion repeats ad infinitum!

1/98 = 0.01020408163265 30612244897 95 9183673 469387755....10204081632.... and repeats ad infinitum.

And:

....102040816326530612244
....897959183673469387755
= 999999999999999999999

√10/98 x 7 = √5.
√10/490 x 7 = 1.
√10/122.5 x 7 = 2.
√10/30.625 x 7 = 4.
√10/7.65625 x 7 = 8.
Etc. etc.
All fractions above are sequences of the decimal expansion above.

And so, one could go on and on endlessly with other rationals.

Circlemaker is on the right track with his 'shadow numbers' i think!

Cheers

PS!
1/1089 gives you the 9 table, and 1/891 gives you the 11 table - but you probably all know this, at least!

reply to post by djeminy


I'm not sure what is so special about the facts you quote that they should not be taught in Universities. We teach them pretty well at the University I am at.

All of the facts are trivialities to do with the decimal expansion of rational numbers. If you google it, you will find it is not a secret, and certainly taught at many Universities.

Or maybe you think that you can patent the actual numbers you have in your list, and then universities won't be able to teach about them.

(As a friend of mine jokes, "there are many copyright and illegal whole numbers".)

Interesting to consider that a line consists of individual points like (1,1) to (1,5), on graph paper. But if you consider that in between each of those points, there can be an infinite number of additional points if you look close enough, then would each of these infinity's add up to be a greater infinity than infinity? Or is infinity simply infinity?

Originally posted by PhysicsAdept
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.

edit on 18-3-2012 by PhysicsAdept because: (no reason given)

To be able to agree to the concept that there are different sized ∞'s, the concept of time must exist. And because we know that time exists there must be different sized ∞'s.

That is if you start with any number, including zero (nothing) and add 1 to it (or any other number, decimal or other) and I start 1 second or 1 millisecond after you as long as you keep adding 1 or any other number and I keep adding that same number I can never catch up to you which shows (not necessarily proves) that at that point in time there are 2 different ∞'s.

But because you can always add 1 to an ever growing number, there really is no such thing as ∞ regardless of what mathematical formulas can and or do show.

[edit to add] in reality there is NO ∞, all there is the concept of ∞.

Originally posted by ClydeFrog42

You see, the idea of this process is represented by limits. As this continues on, as we *approach* infinity (as our number gets larger in larger), the answer approaches zero.

reply to post by WhatAreThey

 

You are making the false assumption that infinity is something that can be "approached". As if you keep dividing forever you will hit infinity and zero at the same time....

You can approach it by simply moving toward it. I just showed you the most simple way I could think of to illustrate that approach but you couldn't even grasp *that*?

How does your brain interpret this, then?:

You are in a room and there is a door on the other side. You can move toward the door, but each time you move, you can only move 1/2 the distance between you and the door.

Imagine having to half the distance an infinite amount of times.

Imagine the door being infinity.

The limit of 1/x as x approaches infinity = 0 doesn't *really* mean it equals zero. It's just the mathematical way of saying that when x gets bigger the answer gets closer and closer to 0. It's saying the *limit* is zero.

It's demonstrably obvious that you've never taken any advanced math courses. If you would actually like to understand what is being discussed, rather than arguing english language definitions, then go take a math course - otherwise, you are just being ignorant.

reply to post by jb1958


Um no, I think there are some infinities that are larger than others, but in this case (at least according to Cantor) those two infinities can be paired and are equal... though common sense would say that there is more in one sat than the other... It is all based on what you believe I suppose.

I never really got into math too much, but I spent a little bit of time with computer programming - which involves a LOT of math...

I like to think outside of the box, and to me I think the real problem is actually with the idea of positive and negative numbers. In basic math we learn that if you subtract more than what you have, you have to go into "negative" numbers - if not, a whole lot more of math would not work - just like the whole divide by zero issue.

It seems to make a lot more sense if you think of there being two seperate zero's (coincidentally, the symbol for infinity looks a lot like a sideways 8, or two zero's hooked together side-by-side at one location). it makes sense that this symbolizes a negative zero connected to a positive zero at a single point, or gateway, or black hole, or whatever you want to call it.

As it turns out, some computer programming already uses both positive and negative zero's to get around certain issues with programming; without doing so there would be a lot of errors all over the place because the computer only understands numerical instructions.

Signed zero is zero with an associated sign. In ordinary arithmetic, −0 = +0 = 0. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero). This occurs in the sign and magnitude and ones' complement signed number representations for integers, and in most floating point number representations. The number 0 is usually encoded as +0, but can be represented by either +0 or −0.
The IEEE 754 standard for floating point arithmetic (presently used by most computers and programming languages that support floating point numbers) requires both +0 and −0. The zeroes can be considered as a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞, division by zero is only undefined for ±0/±0 and ±∞/±∞.
Negatively signed zero echoes the mathematical analysis concept of approaching 0 from below as a one-sided limit, which may be denoted by x → 0−, x → 0−, or x → ↑0. The notation "−0" may be used informally to denote a small negative number that has been rounded to zero. The concept of negative zero also has some theoretical applications in statistical mechanics and other disciplines.

Signed (Positive or Negative) Zero

It's also interesting to me how this was described by the IEEE back in 1985, which is coincidentally the same year that Back to the Future came out


Other cool stuff:

In mathematics, the Riemann sphere (or extended complex plane), named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity. The sphere is the geometric representation of the extended complex numbers C ∪ [∞], which consist of the complex numbers together with a symbol ∞ to represent infinity.
The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0 = ∞ well-behaved. For example, any rational function on the complex plane can be extended to a continuous function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a continuous function whose codomain is the Riemann sphere.
In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P1(C), the projective space of all complex lines in C2. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.

Riemann Sphere

Georg Friedrich Bernhard Riemann [ˈʁiːman] ( listen) (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.

Georg Friedrich Bernhard Riemann

In mathematics (specifically in differential geometry and topology), a smooth manifold is a subset of Euclidean space which is locally the graph of a smooth (perhaps vector-valued) function. A more general topological manifold can be described as a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one-dimensional manifolds, a plane and sphere (the surface of a ball) are two-dimensional manifolds, and so on into high-dimensional space. More formally, every point of an n-dimensional manifold has a neighborhood homeomorphic to an open subset of the n-dimensional space Rn.

Manifold