a(n)=1-10^{-n} is a Cauchy sequence in the reals. The reals are complete, and so the limit of a(n) exists and is a real number. The limit is found using a straightforward exercise in 1st year undergraduate mathematics, and is equal to 1. Hence 0.9r and 1 are the same real number.
Where does the universe end?
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x=0.999...
10x=9.999...
10x - x = 9
9x = 9
x = 1
0.999... = 1
is probably the simplest, but you can prove it through analysis for a more rigorous proof.
edit: as above
Soldato
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Why not?
thats what i thought, read post #113
Why not?
If the big bang theory is correct. There was no matter before it. So the furthest any matter could be is the speed matter was expelled at times the billions of years since the big bang.
Soldato
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No I mean whats wrong with the idea that the universe is expanding, and will continue to do so forever, thus becoming infinitely big? What lies beyond the matter that was ejected at the very beginning of the big bang, that lies at the edge of the finite universe?
It might not continue forever of course, who knows.
It might not continue forever of course, who knows.
0.999r is exactly 1. Trust me I'm a mathematician. taz448's proof is sound.
And the current theory on the fate of the universe is heat death. The universe will expand forever, so all energy in the universe will be distributed throughout a greater and greater volume, thus the energy density of the universe will tend to 0.
And the current theory on the fate of the universe is heat death. The universe will expand forever, so all energy in the universe will be distributed throughout a greater and greater volume, thus the energy density of the universe will tend to 0.
Speak for yourself. I can comprehend the infinite perfectly fine thank you very much.
You can't have a fraction of infinity, because the fractions would still be infinite. If their was infinite energy in the universe then that is all their would be everywhere, we wouldn't exist.
The average density of the universe today is only about 2-3 hydrogen atoms per cubic metre, when you consider the number of atoms in a planet or star this is quite astonishing, and goes to show just how empty the universe is. Soon after the big bang the universe was so dense that it was opaque, we can't observe anything before this time because any light would have been absorbed and re emitted or scattered.
The average density of the universe today is only about 2-3 hydrogen atoms per cubic metre, when you consider the number of atoms in a planet or star this is quite astonishing, and goes to show just how empty the universe is. Soon after the big bang the universe was so dense that it was opaque, we can't observe anything before this time because any light would have been absorbed and re emitted or scattered.
Assuming the logic behind curved space is correct, and locally to our planet, it would appear to be. And also assuming that the big bang theory is correct, which at the moment it would appear to be at least roughly true, then it makes sense what we call "the universe" would be roughly spherical in shape.
Using this logic, it is fair to say that the universe is probably not infinite, in the same way that the planet is not infinite, it's just that we are so incredibly tiny in the scheme of things, given a large enough sphere, things take on a finite planar form to our perception.
In my opinion, in the same way as if you started at one point on the Earths surface and wandered all the way around, you would end back at the same place and be able to measure a definate value for its circumference, if you moved away from the Earths surface by a value of R (radius) and managed to then circumnavigate space in such a way that you always maintained R from the Earth, you would eventually end up back where you started in space and again would have a definate value for the circumference of the imaginary sphere you have just gone around in space (which you would hope would be proprtional to the value of the Earths Radius + R.)
Given all of this and assuming it is true, if you were then able to move away from the Earth at a speed exceeding the expansion of the universe (whatever that may be, this is where my physics lets me down though I assume it will be to do with traditional energy properties such as inertia etc.) then in theory you would evetually exceed the boundaries of the universe. HOWEVER in my personal opinion, the definition of our universe is matter and energy.
Space, by definition is not matter or energy, those things are what exist in place of space (existance by negation effectively).
Therefore IF one was to exceed the boundaries of our universe, what you would actually experience would be space, no matter, no energy, just space.
Therefore, space is infinite, but the universe is not, assuming a fixed point in time (given it is ever expanding or contracting, changing either way).
Or at least, that's how it all makes sense to me!
Using this logic, it is fair to say that the universe is probably not infinite, in the same way that the planet is not infinite, it's just that we are so incredibly tiny in the scheme of things, given a large enough sphere, things take on a finite planar form to our perception.
In my opinion, in the same way as if you started at one point on the Earths surface and wandered all the way around, you would end back at the same place and be able to measure a definate value for its circumference, if you moved away from the Earths surface by a value of R (radius) and managed to then circumnavigate space in such a way that you always maintained R from the Earth, you would eventually end up back where you started in space and again would have a definate value for the circumference of the imaginary sphere you have just gone around in space (which you would hope would be proprtional to the value of the Earths Radius + R.)
Given all of this and assuming it is true, if you were then able to move away from the Earth at a speed exceeding the expansion of the universe (whatever that may be, this is where my physics lets me down though I assume it will be to do with traditional energy properties such as inertia etc.) then in theory you would evetually exceed the boundaries of the universe. HOWEVER in my personal opinion, the definition of our universe is matter and energy.
Space, by definition is not matter or energy, those things are what exist in place of space (existance by negation effectively).
Therefore IF one was to exceed the boundaries of our universe, what you would actually experience would be space, no matter, no energy, just space.
Therefore, space is infinite, but the universe is not, assuming a fixed point in time (given it is ever expanding or contracting, changing either way).
Or at least, that's how it all makes sense to me!
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Soldato
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1/3 is more than 0.33r. Like I said before, a limitation of the decimal number system, you cannot divide 10 by 3 and get a practical number.
If you were to use the example:
X = 0.9r
10X = 9.9r
10X = X = 9 = 9X
X = 1 = 0.9r
First, you must apply a limit to the recurring digit somewhere. For calculation's sake, I'll use 3 digits, but it applies to any number of digits.
X = 0.999
10X = 9.999 - That cannot be right, because now you have multiplied 0.999 by MORE than 10 to get 9.999, thus, it is not 10X.
So 10X = 9.990
10X - X = 9.990 - 0.999 = 8.991 = 9X
8.991 / 9 = 0.999, but not 1.
If you did not apply a limit, either you would be multiplying X by more than 10 (disproving the theory), or X would refer to two numbers, which is impossible.
a(n)=1-10^{-n}
I have no idea about that.
Also WRT to the argument about whether 0.99r = 1, its an interesting question but the answer lies in the impossibility of discritising the infinate. Number systems will always be inadequate, that is their nature. As I understand it, 0.99r is considered to be the same as 1 by most. As has been posted here, it is possible to form cognitive proofs as to why this should be so, however it is is still using two different labels to represent the same thing.. sometimes you have to wonder if that seems right?
Number systems and maths in general are an amazing thing, its easy to argue that maths is possibly one of mankinds greatest achievements. However given we utilise something (be that space on paper, magnetic patches on hard disks etc.) to represent a discrete portion of an infinate selection, there can never be enough matter or energy at our disposal to represent the infinate.
That's the nature of infinity, and lets face it, everything we deal with is infinite. Whether it be harnessing energy or exploring matter, there will always be something about it that is infinite and therefore we can never express everything about it.
I think that's where infinity causes most of its problems, we don't like the idea that there is something we simply cannot solve, cannot do, but infinity is that one thing we can never master as by its very definition, it can never end. In fact you could argue that simply by conceiving of the infinite, giving it a name and describing it, we have mastered it.
At least in traditional discrete mathematics, high discrete precision is rarely a huge issue when dealing with actual values... think of us poor computer programmers, with our effective x10^10 limit on single precision floating point accuracy, especially those of us who try and use such woeful limitations to make numerical simulations
Number systems and maths in general are an amazing thing, its easy to argue that maths is possibly one of mankinds greatest achievements. However given we utilise something (be that space on paper, magnetic patches on hard disks etc.) to represent a discrete portion of an infinate selection, there can never be enough matter or energy at our disposal to represent the infinate.
That's the nature of infinity, and lets face it, everything we deal with is infinite. Whether it be harnessing energy or exploring matter, there will always be something about it that is infinite and therefore we can never express everything about it.
I think that's where infinity causes most of its problems, we don't like the idea that there is something we simply cannot solve, cannot do, but infinity is that one thing we can never master as by its very definition, it can never end. In fact you could argue that simply by conceiving of the infinite, giving it a name and describing it, we have mastered it.
At least in traditional discrete mathematics, high discrete precision is rarely a huge issue when dealing with actual values... think of us poor computer programmers, with our effective x10^10 limit on single precision floating point accuracy, especially those of us who try and use such woeful limitations to make numerical simulations
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I don't see why people have a problem with infinite space yet would be open to the idea of the multiverse existing in an infinite number of possible dimensions, surely having one space is more logical and likely?
If you have one infinite expanse of space where big bang events can happen but do so at unimaginable distances from each other, you still get the multiverse in a way but you'll never see them as they would be so rare and far from each other, i mean think about it, where exactly is here and why would there be other dimensions if space is infinite as it likely is?
I've posted this before in a few threads dealing with this sort of thing, makes for a very good read on infinity in time and space.
If you have one infinite expanse of space where big bang events can happen but do so at unimaginable distances from each other, you still get the multiverse in a way but you'll never see them as they would be so rare and far from each other, i mean think about it, where exactly is here and why would there be other dimensions if space is infinite as it likely is?
I've posted this before in a few threads dealing with this sort of thing, makes for a very good read on infinity in time and space.
Rubbish. You divide 10 by 3 and get 10/3, which when represented in base ten, is written as 3.3r. The number represented by an infinite decimal expansion is defined as the limit (mathematical term, precisely defined) of a sequence of partial sums. In this case:
0.3r =(definition) lim_{n->infinity} 3( 1/10 + 1/100 + 1/1000 + ... + 1/10^n ) =(use definition of limit) 1/3
There's no ifs and buts. All this is perfectly well defined, and very straightforward.
1/3 is more than 0.33r. Like I said before, a limitation of the decimal number system, you cannot divide 10 by 3 and get a practical number.
If you were to use the example:
X = 0.9r
10X = 9.9r
10X = X = 9 = 9X
X = 1 = 0.9r
First, you must apply a limit to the recurring digit somewhere. For calculation's sake, I'll use 3 digits, but it applies to any number of digits.
X = 0.999
10X = 9.999 - That cannot be right, because now you have multiplied 0.999 by MORE than 10 to get 9.999, thus, it is not 10X.
So 10X = 9.990
10X - X = 9.990 - 0.999 = 8.991 = 9X
8.991 / 9 = 0.999, but not 1.
If you did not apply a limit, either you would be multiplying X by more than 10 (disproving the theory), or X would refer to two numbers, which is impossible.
a(n)=1-10^{-n}
I have no idea about that.
Do you enjoy being wrong, or are you just foolish?
You're applying a limit on the number of decimal places. There is no such limit, that's why it's called recurring. So when you multiply it by 10, the n'th decimal place is still 9 for all n. It's really not that difficult to work out.
That's how I have always assumed things would work.
The infinate is space and given we have a fixed point in space ourselves, we are even able to apply geometry to working out where things lie.
in my very humble experience, systems naturally tend to lead to larger systems which resemble their smaller components.
Atoms to Matter
Matter to Planets
Planets to Solar Systems
Solar Systems to Galaxies
Galaxies to the Universe
So, given space is nothing and therefore by definition infinate, why not :
Universes to ??? (profit?)
When dealing with infinity, such as space has to be (unless of course mankinds traditional God structure is in place, in which case things probably end somewhere after orions belt in reality) Then why does the universe have to be the ultimate? Why cant there be multiple universes making up something else much larger, and then multiple instances of the larger thing making up something else? I'm not sure conceptually anybody should ever say that the universe is as big as it gets until they have proved that to be true.
However given how teeny tiny we are I also think it unlikely we will be abble to provide such a proof before our species as we know it becomes extinct for whatever reason... Very much like God, it seems as though it is something that we can never really prove. Still at least it all has a footing in testable reality!
Having said that, things like the String theory do complicate matters somewhat, though I do sometimes wonder if trying to solve the complex intricacies seen at the very small sub-atomic level that then bear fruit to much more far reaching all encompassing theories that try to redefine reality as we know it and experience it, are perhaps sometimes misguided and simply too grand for their own good! Sure, very odd things happen the smaller we go, and explaining them often leads to very radical theories, but we simply cannot dismiss our own experiential evidence of the universe around us either...
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