Maths Question Regarding Infinity

[Fanta Grape]

[Thread has been answered]


So I was thinking about the concepts behind infinity and I found a very strange problem which I can't seem to fix that uses quite basic maths. Here we go and hopefully you can tell me where I went wrong.
An ellipsis will represent recurring numbers.
INF = Infinitely Large Number

1/3 = 0.3...
[Multiply both sides by 3]
3/3 = 1 = 0.9...

[Alternatively]

Let x = 0.9...
[Multiply both sides by 10]
10x = 9.9...
[Remove x from both sides]
9x = 9
[Divide both sides by 9]
x = 1
1 = 0.9...

1 - 1 = 0
1 - 0.9... = 0.0...1

1 = 0.9, Therefore, 0 = 0.0...1 *

0.0...1 = 1/INF
0 = 1/INF
[Multiply both sides by an INF equal to the previous INF]
0 = 1 **

*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.

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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?

---

tl;dr

I made a boo boo and I need to find it. Help.

 

[Batou667]

1/9 = 0.111111111...

Therefore 9/9 = 0.99999999....

But shouldn't 9/9 = 1?

Yeah, it's a well-known maths paradox/fallacy. Don't worry about it too much.

 

[generals3]

Actually the reason why 0.33333... * 3 = 1 is because in Real Numbers infinitesimals don't exist so you round it up. If you allow infinitesimals to exist than 1/3 =/= 0.333.. to begin with. (unless i'm mistaking here) (i think that if you want to get to the bottom of this you'll need to look into Hyperreal numbers which allow the existence of "infinitesimals" and "infinite")

 

[DasDestroyer]

It's the old "Does 1=0.9999999..." question. The basic answer(and pretty much the only one I can give you) is that since this deals with infinity, normal laws do not completely apply.
If you were to round the numbers ever so slightly, just so that they aren't infinitely long, you get:
x=0.9999
10x=9.9990
9x=8.9991
x=0.9999

 

[SckizoBoy]

I made a boo boo and I need to find it. Help.

Easy... definition.

0.000...1 can't be defined, not in the way you've expressed it (or any way unless a college maths student wants to correct me).

Problem A - For the analogy you've given (time) you've ballsed up the context. One hour in the context of infinity may be an infinitesimally short amount of time, but to an astronomer, a hundred years is a short amount of time, but to a particle physicist, a femtosecond is a hideously long time frame. Moreover, any number that has mathematical definition will divide into infinity to give a result of 0. By its own (lexicographical) definition, infinity has no (mathematical) definition.

Problem B - Against the above, multiplying anything by infinity as a mathematical function will leave you with a headache. Since it has no definition, the result cannot be defined either.

 

[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.

Edit: Yes, there are different kinds of infinities.
Most popular Example: There infinite natural numbers. But they are countable.
There also are infinite real numbers. Those are not countable.
In Math-language: The cardinality of N is smaller than the one of R.

 

[Who Dares Wins]

ITT: People confusing approximately equal (&#8776 with equal (=).

Your theory was proven wrong when you posted "1/3 = 0.3..."

 

[ManOwaRrior]

ITT: People confusing approximately equal (&#8776 with equal (=).

Your theory was proven wrong when you posted "1/3 = 0.3..."

But it is. Once you understand what 0.3... means, you will understand why.
(Crosses fingers for a page-long, totally nonsense, yet still funny debate).

 

[Simmo8591]

Your problem is that you are too hung up on infinity just being a really big number but it's a concept that we use to show when there will always be a set of numbers that will exist.

in the case of 1/Inf which you mentioned we commonly use epsilon to denote a very very very (infinitely) small positive number, we then see if such a tiny number can be squeezed into small gaps between numbers, whether it can or cant helps us define things like left and right continuous functions, upper and lower bounds and other horrible complex stuff that I cant define properly off the top of my head...

tl;dr

It's not a real number its a way of saying there will always be a possible number

 

[the spud]

Ah, a variation of the old "Does .999999999=1" thread. According to Cracked, this is one of those topics that sound innocent, but somehow provoke flamewars.

 

[Valagetti]

The entire whats a thrid of 10=3.3... pisses me off too, people do write about it and if you google scholar it, you'll get some interesting papers on the subject.

 

[Johnson McGee]

I'm pretty sure you can't cancel 1/Inf by multiplying by Inf, you would just get Inf/Inf which is undefined.

 

[Jaksteri]

Resulotion to that, allways round up or down. Not the most accurate form but seriously, who expects things to be more accuate than they need to be. Just because my work I often come face to face with "infinite" numbers, the thing is I don't have machinery that would operate with that high acurazy, thus rounding them to most accurate number of desimals I can actually work with is the best and only option.

No need to make things harder than they are you only end up with migrein. As mentioned earlier 1/3 to make 3/3 ever 1 one of the 1/3 needs to "end" with 4 to ever make 100%, this would however imply that one of the equal parts would not be egual to others.

 

[Fanta Grape]

Actually the reason why 0.33333... * 3 = 1 is because in Rational Numbers infinitesimals don't exist so you round it up. If you allow infinitesimals to exist than 1/3 =/= 0.333.. to begin with. (unless i'm mistaking here) (i think that if you want to get to the bottom of this you'll need to look into Hyperreal numbers which allow the existence of "infinitesimals" and "infinite")

Actually the reason why 0.33333... * 3 = 1 is because in Rational Numbers infinitesimals don't exist so you round it up. If you allow infinitesimals to exist than 1/3 =/= 0.333.. to begin with. (unless i'm mistaking here) (i think that if you want to get to the bottom of this you'll need to look into Hyperreal numbers which allow the existence of "infinitesimals" and "infinite")

ITT: People confusing approximately equal (&#8776 with equal (=).

Your theory was proven wrong when you posted "1/3 = 0.3..."

The entire whats a thrid of 10=3.3... pisses me off too, people do write about it and if you google scholar it, you'll get some interesting papers on the subject.

That's why I provided my alternate route to get the same result. Didn't anyone see it? It's right below the word [Alternatively]

I already realised that 0.3... might not equal 1/3, so I posted something that uses a different sort of logic I used in high school.

Although as for the rest of the responses, would you mind explaining how definitions work into this? I read "there's no definition for a number," but I'm not quite sure what it means.

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.

 

[DefunctTheory]

Ugh. This problem.

Remember this - Numbers and math are not REAL. Their human creations to quantify and calculate real things. And since math is not real, it cannot truly and accurately answer every question.

Until, of course, you discover the greatest mathematical tool I have ever encountered. The term '-ish.'

As in 1=.999...ish.

Bam. Problem Solved.

 

[LuminaryJanitor]

where did I go wrong?

You're putting things that aren't numbers into equations. 1/INF is not a number, and neither is INF. 0.000...1 is not a number; it's a meaningless piece of notation you invented yourself. "Infinitely small numbers" don't exist. If it's a number, then I can name a smaller one.

 

[Robert632]

snip

Spoiler: This is you, intentionally or not.

O.t: Yeah...this is just one of those things that happens in math. At this point, most people try not to talk about it.

 

[GoAwayVifs]

As in 1=.999...ish.

No. No, no, no no, no. 1 is exactly equal 0.9 repeating. I know that it is a bit hard to warp your head around but it is true. Wikipedia has a full article on it that is quite informative.

In regards to the OP, infinity is not a number so any operations you do with it are meaningless, they have no answer.

 

[generals3]

That's why I provided my alternate route to get the same result. Didn't anyone see it? It's right below the word [Alternatively]

I already realised that 0.3... might not equal 1/3, so I posted something that uses a different sort of logic I used in high school.

Although as for the rest of the responses, would you mind explaining how definitions work into this? I read "there's no definition for a number," but I'm not quite sure what it means.

Again it's a matter of which system you use. Real numbers (most used system (though the imaginary one is often used as well) by the "common men") doesn't allow the existence of infinitesimals. Whenever you try to get to an infinitesimal in R you will get "0" so 0.99999...+0 =1. 1/infinite means nothing in Real Numbers considering "infinite" doesn't belong to R (that's why when computing limits you often use R Union (-Infinite, +Infinite)).
And if you go beyond R into hyperreal numbers than 1/3 =/= 0.3333... That simple.

(unless i'm mistaking)

 

[The_root_of_all_evil]

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.