Maths Question Regarding Infinity

[ManOwaRrior]

0.000...1 is indeed not defined in mathematics.
If one would try to define it, he would find that 0.00...1 = 0 = 0.00...2 = 0.00...9.
(Can prove if needed. For now, just realize that 0.00...1 - 0 has to be smaller than any given positive number and can't be negative, thus it has to be 0. From there it's just a-b=0=>a=b).

Problem A: Time, as we perceive it, is not infinite. That's one Point where your Problem falls apart. The other Point is that 1/infinity is not defined. It is not defined, because infinity is not a number. If it was, 1/infinity had to indeed be zero (same proof as above), but then we'd have 1/inf = 0 => 0*inf = 1. And that's not making any sense.

Problem B: Evaporates once you realize that the Term 1/inf is not defined.
Second question in B: The smallest possible decimal Number is 0. If you want the smallest possible positive decimal Number, well, it doesn't exist.
Google the concept of an open set to learn why. Easy argument: For every positive Number x, no matter how small, there is an even smaller one, x/2 for example, that is still positive.

Your first proof makes sense, but that would simply bring me to the conclusion that 0.0...1 = 0, and then therefore, 0.9... = 1, despite my alternative proof. Would you care to explain that? (That came out a bit sarcastically, but I'm quite sincere, believe me). [Edit: Bleh, misread that AND articulated the response incorrectly. Could you explain how definitions work into this?]

Also, you really just answered problem A by restating the question. I stated that INF is just something I used to express as an "infinitely large number". Obviously 1 cannot equal 0 so where did I go wrong?

Regarding the smallest possible positive number, I know it doesn't exist. My issue was that would all infinitely small numbers be the same? If they were, then it could be stated that 1/INF = 0.0...1.

Definitions work into this because all of mathematics rely on properly defined terms.
We can write stuff like 1 + 1 = 2, because 1,2,+,and= are all defined things. There are explicit rules how to use them.
Your problems arise once you use terms like "an infinitively large/small number".
Those terms, as you use them, are not defined in mathematics and can therefore not be used in a mathematic context.
Your alternative proof that 1=0 falls apart when you claim that 1/inf * inf = 1.
There are no defined rules in mathematics, how to divide and multiply with "infinitively large/small numbers", precisely because you run into problems like this.

Those numbers and rules exist in the hyperreals, but they exceed my mathematical understanding.

 

[Cryptocyanid]

Also, you really just answered problem A by restating the question. I stated that INF is just something I used to express as an "infinitely large number". Obviously 1 cannot equal 0 so where did I go wrong?

The 1/Inf you are using is essentially shorthand for [ 1/x ] for x approaching infinity.

Now to keep this relatively short, and not entering the pointless debate on weather .9999.. is 1, the place you went wrong is where you multiplied both sides of an equation by your infinitely large number.

You cannot do that.

Whether you are using Real or Hyperreal numbers you can only multiply both sides of an equation with a number (or variable representing a single number) not an undefined quantity.

Alarm bells should be ringing when you state that 'it is the same infinitely large number' as this really is not covered by the Real numbers, as you could easily chose to have another infinitely large number that is smaller or larger if it could also be the same, which in the context of Real numbers makes no sense.

Hyperreal numbers permit algebraic operations with infinities and infinitesimals, however, they don't map 1:1 to Real numbers so reasoning about them intuitively becomes difficult, and no practical use for them has been found yet as far as I know. In practice having an infinite numbers of values that all map to Real zero but are different Hyperreal numbers is impractical.

 

[Fanta Grape]

Also, you really just answered problem A by restating the question. I stated that INF is just something I used to express as an "infinitely large number". Obviously 1 cannot equal 0 so where did I go wrong?

The 1/Inf you are using is essentially shorthand for [ 1/x ] for x approaching infinity.

Now to keep this relatively short, and not entering the pointless debate on weather .9999.. is 1, the place you went wrong is where you multiplied both sides of an equation by your infinitely large number.

You cannot do that.

Whether you are using Real or Hyperreal numbers you can only multiply both sides of an equation with a number (or variable representing a single number) not an undefined quantity.

Alarm bells should be ringing when you state that 'it is the same infinitely large number' as this really is not covered by the Real numbers, as you could easily chose to have another infinitely large number that is smaller or larger if it could also be the same, which in the context of Real numbers makes no sense.

Hyperreal numbers permit algebraic operations with infinities and infinitesimals, however, they don't map 1:1 to Real numbers so reasoning about them intuitively becomes difficult, and no practical use for them has been found yet as far as I know. In practice having an infinite numbers of values that all map to Real zero but are different Hyperreal numbers is impractical.

Okay, thanks a lot. I understood your response the best sense to me. I appreciate that you took the effort to make an account just to respond to me, or at least I was your first thread by coincidence. Baha.

Oh dear god, not this one again.

Infinity cannot be represented by finite operations, so you can create a whole slew of paradoxes by treating it as such.

Same reason you can't divide through by zero.

For a start, Infinity cannot equal Infinity, because equals is a finite operation.

It's even referenced in HitchHikers.

It is known that there is an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you meet from time to time are merely the products of a deranged imagination.

If you want to know where you went wrong, it's that recurring numbers are an approximation, and it's measuring error that's causing the paradox. You can't say "equals" with an approximation.

Thanks. Planned on actually getting the hitchhiker's books for Christmas =P
Makes sense and I appreciate that you took the time to respond.

 

[])rStrangelove]

*Problem A:

0.0...1 is an infinitely small number. Imagine it as 1 divided by an infinitely large number. Now hypothetically, let's say that time is infinite, for argument's sake. One hour out of an infinite amount of hours would then equal 0. That doesn't make logical sense.

**Problem B:

Zero doesn't equal one ... And we can multiply both sides by whatever number we like to make even more nonsense.

---

Now there's a few questions raised here. First of all, can there be different sizes of infinitely small and infinitely large numbers. Some may argue that 0.0...1 does NOT equal 1/INF. This makes sense in the example of the hypothetical time, where 3 hours is longer than 2 hours. While both infinitely small, one is larger than the other. Although this raises the question as to whether it would actually make a difference in the calculation. Also, let me propose another calamity. If I were to ask you of the smallest possible decimal number, could you not multiply that by 10 and it wouldn't get any larger as it can always get smaller?

Its a theory, nothing more. It doesnt appear to make sense because whatever you do with something incredibly small/huge is totally meaningless because of its massive scale.

10 x INV <= wouldnt change anything, INV is so big it doesnt matter, its still INV

The point is not to have a clear result by using this theory. We use it to see how things change when something slowly becomes incredibly huge or small. Thats all.

 

[Subbies]

INF/INF = Undef.

 

[TheIronRuler]

You have a line in space, it exists in two dimensions. It has an infinite amount of points on it.
You have a plain in space, it exists in three dimensions. It has an infinite amount of points on it.
Which one has more points?
Theoretically, the infinite amount of points in the plain is more than the infinite amount of points in the line.
How can - INFINITE A>INFINITE B?
.
My answer - Drop the issue.

 

[The_root_of_all_evil]

Thanks. Planned on actually getting the hitchhiker's books for Christmas =P
Makes sense and I appreciate that you took the time to respond.

Sorry it was a little terse, but there have been huge flame wars fought over that in the Escapist past. Good to see there's been some edumacation since then

 

[Haelium]

Oh great, one of these threads where people who actually know very little about maths use big words and walls of text to argue about something within maths that really does not matter.

It's pretty fucking simple, 0.33333... is not a problem with maths, just our numerical system, decimals like that just don't work, it's that simple. Stick to fractions and you won't get these problems. There is no paradox, just an error in our representation of numbers.

 

[LuminaryJanitor]

You have a line in space, it exists in two dimensions. It has an infinite amount of points on it.
You have a plain in space, it exists in three dimensions. It has an infinite amount of points on it.
Which one has more points?
Theoretically, the infinite amount of points in the plain is more than the infinite amount of points in the line.

No, theoretically, they're exactly the same. And they're both the same as the number of points in (0, 0.000000001).

Similarly, the number of integers is equal to the number of rationals (fractions). But they're both strictly less than the number of points in the real line.

 

[Nalgas D. Lemur]

(Crosses fingers for a page-long, totally nonsense, yet still funny debate).

It somehow never stops being entertaining. I don't know that dropping words like "countable" and "cardinality" in the previous post will mean anything to anyone who doesn't already get it, but at least it's something for people to look up. Heh.

Theoretically, the infinite amount of points in the plain is more than the infinite amount of points in the line.
How can - INFINITE A>INFINITE B?

As has already been pointed out, both of those are the same, even though they at first may seem not to be. A good place to start trying to understand how things like that work and also how two different infinities can be different is to look up countable/uncountable and find a good explanation of Cantor's diagonal argument. The one on Wikipedia is unnecessarily hard to understand, just like most of their math stuff, unfortunately, and I need to go get something to eat and don't have time to look for something better right now.

 

[Zantos]

When you do real analysis you start everything off with "Define an arbitrary positive epsilon <<1" To avoid things like this. I miss my arbitrary positive epsilon << 1.

 

[isometry]

Yeah 1 = 0.9999... , it's just two ways of writing the same number. There is no fallacy or paradox, just two labels for the same number.

If you want to know where you went wrong, it's that recurring numbers are an approximation, and it's measuring error that's causing the paradox. You can't say "equals" with an approximation.

This is wrong. Recurring decimals are not an approximation unless you terminate them. The equation:

1/3 = 0.333...

is exact and rigorous. The right-hand side is short hand for an infinite series:

0.333... = 3/10 + 3/100 + 3/1000 ... = Sum(3/(10^n), n from 1 to infinity).

You may have seen infinite sums like this before in math classes. They are rigorously defined as long as they converge. It's also true that if an infinite series converges then we can multiply it by a constant, so multiplying by 3 is rigorous:

1 = 0.999...

This sum is exact and rigorous. It gives us two different labels for the same number. There is no paradox.

 

[BehattedWanderer]

Ugh, you mathematicians and your desire to prove things. This is why I went engineering, they quickly realized that spending time worrying over itsy bitsy things like this was time not spent actually addressing the issue.

Measure as finely as you care to for practical purposes, and slap a +/- 1/2[smallest measuring unit in use] on the results.

Bam. Done. Write up the bill, and move along. Incorporating practical mathematics and theoretical mathematics can be done, but not without some small rule bending. Approximate is not equal, suffice it to say, and add a small variation at the end to account for what may have been missed, or what may have been accidentally (or intentionally) added, and bam. Moving right along.