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2=1.999...infinity! What are your thoughts?
- Thread starter sjjordan
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clayj said:
There is no last nine. The difference is zero. They are two ways of writing the same quantity. If you want to claim otherwise, you need to produce a real number that lies between the two, and no such number exists.
Well, I only included a terminal 9 in order to make the number look nicer...Santiago said:
I realize that in an infinite number of 9's, there is no last 9.But surely it's obvious that for ANY number "x", x != x-1?:
1 != 0
2 != 1
3 != 2
...
infinity != infinity - 1
If you multiply both sides of the equation
1.99999999... = 2
by 10000000... (1 followed by infinity zeros), you'd get
199999999... = 200000000...
and that's wrong, too.
Well, of course... you're dealing with matter and energy, neither of which exist in or deal with infinities.jsw said:
From a purely mathematical perspective, however, I would have to say that they are NOT equal. It doesn't matter if you can't find a real number between "x" and "y"... they're not the same. The difference may be infinitely small, but that's OK because we're talking about decimals that stretch to infinity. Besides, infinity's not a real number (in that it cannot be expressed using digits) anyway... it's an abstract concept that has no logical basis in this universe.
Actually, infinities do exist in the world of physics, for example when dealing with black holes, which are real... pending, of course, certain m-brane theories to the contrary.clayj said:
But, agreed, for the most part, it's an abstract concept.
OK, leaving God out of it, there's not sufficient computational power available in the entire Universe to express infinity.appleretailguy said:
The Bekenstein Bound sets an absolute limit on the amount of information that can be stored within a given region of space... so, even if we had godlike power and could harness all of the matter and energy within the Universe to computational purposes, the fact that the Universe has a finite size dictates that we would NOT have infinite computational power.
How do we know the Universe is not infinite in size? Easy. The sky is black at night. An infinite Universe would have stars in EVERY possible direction, so the sky would be white at night. A black sky means the Universe is finite.
So, if we had godlike power, we could talk about virtual infinitude (defined as "infinite for all PRACTICAL purposes"), but true infinity will ALWAYS be beyond our reach.
1.999infinity=2
Theres a basic reason to this
If you can not seperate 2 numbers by another number then the numbers by definition have to be equal since there is never a dividing point between the numbers. Sorry for the laymans tallk but im tired. I think the biggest problem is that people cant handle the concept of infinity. I mean take the concept of Fourier Series. This pretty much lets you rewrite any perodic function as a combination of sines and cosines...... and when taken to infinity, it WILL become that function. Same idea applies here. People just need to become more knowledgeable about infinity. The concept of infinite series is pretty much standard curriculum in any calculus 2 course.
Night,
Jonathan
Theres a basic reason to this
If you can not seperate 2 numbers by another number then the numbers by definition have to be equal since there is never a dividing point between the numbers. Sorry for the laymans tallk but im tired. I think the biggest problem is that people cant handle the concept of infinity. I mean take the concept of Fourier Series. This pretty much lets you rewrite any perodic function as a combination of sines and cosines...... and when taken to infinity, it WILL become that function. Same idea applies here. People just need to become more knowledgeable about infinity. The concept of infinite series is pretty much standard curriculum in any calculus 2 course.
Night,
Jonathan
Not true.clayj said:
Expansion theories postulate that the universe expanded faster than the speed of light for a period, meaning light from the most distant stars has not and may never reach us. Given sufficient time, current theory is that the universe will be bathed in light.
Also, there is no basis for the argument that the universe is finite, especially given current 11-dimensional m-brane/string theory, which provides lots of places to store infinite space.
Where did you read THAT? I've read Brian Greene's books, and I've never heard of any supraluminal expansion period.jsw said:
And the Universe is already bathed in the 3° K background radiation from the Big Bang...
OK, theory. I'm more interested in what's been scientifically proven... the farthest distant known object at 13.5 billion light-years (forgive me if the figure is slightly off), the known rate at which it's receding (based on red shift), etc. Theory may postulate an infinite space packed into extra dimensions, but what we know empirically is that the Universe is finite...jsw said:
Anyway, I still think that 1.99999999.... != 2.
The problem, of course, is that our understanding of the Universe is not what we'd like for it to be.appleretailguy said:
But the thing is, if we assume that all of spacetime is behaving similarly, then the fact that objects 13 billion light-years away are receding at 97% of the speed of light implies that the Universe CAN'T be much larger than that, because no object could be so far away that its speed of recession is greater than c (unless, of course, Einstein was wrong... and I ain't going there). Ergo, the Universe CAN'T be infinite, UNLESS it's not all expanding. But everywhere we look, we see signs of expansion.
Occam's Razor, then, says that the Universe is finite. Ergo, we cannot actually express infinity, except as an abstract (albeit useful) mathematical concept.
ok, time to think about this logically
the difference between 2 and 1.99... is a 1 preceded by an infinate number of zeros, infinite means never ending, so the zeros never end, thus the 1 never comes, so if the difference is an infinate number of zeros, and nothing but zeros(hence infinite), then the difference is zero.
there is no difference
the difference between 2 and 1.99... is a 1 preceded by an infinate number of zeros, infinite means never ending, so the zeros never end, thus the 1 never comes, so if the difference is an infinate number of zeros, and nothing but zeros(hence infinite), then the difference is zero.
there is no difference
They are different, but only in theory. The moment you start trying to do summat with it, it becomes 2, because the difference is that small, and it's easier to work with
Haha they are not different in theory thats the whole pt of this arguement which isn't much of an arguement to begin with.
Jonathan
Jonathan
They are different. It may be 0.00r1, but they're still different, and that *could* make a difference to certain parts of theoretical physicsdukebound85 said:
greatdevourer said:
ok, lets take a different example, .333... is a rational number, it is equal to 1/3, and as far as i know, no one has ever disputed that, so, .333...x3 equals .999... whereas 1/3x3 equals 1
if you take two equal numbers and do the same things to them, the answers will be the same, meaning .999... is equal to 1
like i said before, the difference would be a 1 after infinite zeros, and since the zeros go on to infinity before stopping, thus they don't ever stop, thus the 1 never comes; the difference is just alot of zeros, so there is no difference, not logically, not mathmatically, not theroetically, not physically, and i'd like to see someone try to find some theoretical equation where that wasn't so
The mistake in this argument is that 1 followed by infinity zeros is an infinitely large number, in other words infinity (more precisely, Aleph-0), and Aleph-0 does not share arithmetic properties with real numbers such as 1.99... and 2. For example, Aleph-0 is equal to one less than itself and knowing that Aleph-0 times A is equal to Aleph-0 times B does not imply that A=B.clayj said:
Your values 199999999... and 200000000... are also infinite, so you can't call them unequal based on integer properties.
The reason that 1.99... and 2 are equal is that there is no value between them. They are unequal if and only if there is a value between them (which you might find by averaging). But that's not the case, for the following reason:
Suppose N exists with 1.99... < N < 2. Being less than 2 means it must begin with 1 and some decimal digits. If any one of those digits is other than 9, then N=1.999...D... for some digit D, but then N=1.999...D... < 1.999...9... because D<9, so N is not greater than 1.99... (contradiction). Therefore, every decimal digit of N must be 9, so N=1.99... and again N is not greater than 1.99... (contradiction). Since our supposition leads only to contradictions, the claim that N exists is false.
I love the song they quote:
"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall".
I remember the first time I ran into this problem. 8th grade. We were pounding through conversion of decimal to rational numbers.
I got the answer. I didn't like it. I checked the teachers book and still didn't like it. It was like someone killed Santa and stoned the Easter Bunny.
Then I sat down and did the 1/3 +2/3 and I saw the hole. That the numbers were overly complete and there were some times when more than one number was the same thing.
Santa was still dead but the world made sense.
Since then I've been much more comfortable with numbers and yes, as an engineer I have rounded pi off to 10.
I got the answer. I didn't like it. I checked the teachers book and still didn't like it. It was like someone killed Santa and stoned the Easter Bunny.
Then I sat down and did the 1/3 +2/3 and I saw the hole. That the numbers were overly complete and there were some times when more than one number was the same thing.
Santa was still dead but the world made sense.
Since then I've been much more comfortable with numbers and yes, as an engineer I have rounded pi off to 10.
MongoTheGeek said:
That seems like some awfully generous rounding! 3.1415...all the way up to 10???
Hahaha.(I'm assuming...hoping...for our safety and wellbeing, you mean decimal places!)
tobefirst said:That seems like some awfully generous rounding! 3.1415...all the way up to 10???
(I'm assuming...hoping...for our safety and wellbeing, you mean decimal places!)
Nope. I rounded it up to 10. I was just interested in the order of magnitude. It was a quick and dirty calculation to do wind load and see if something was sturdy enough. If the answer was no or maybe I would actually do the math. As was, the answer came back 100m/s and my roundings were in the safe direction so I said screw it.