How the hell can INFINITY be larger than another INFINITY?! ITS UNLIMITED! IT NEVER STOPS! ITS GOD DAMN FUCKING HUGE!!! *Brain Explosion*corroded said:
Infinity = 1?
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1 is a defined digit and bedrock for all of mathematics. It's also a concept.
Infinity, however, is purely a concept. It's endless, and utterly impossible to imagine. It's not "all of X thing", as that would still be a number. Trying to think of infinity is futile, much like imagining the 4th dimension.
Infinity, however, is purely a concept. It's endless, and utterly impossible to imagine. It's not "all of X thing", as that would still be a number. Trying to think of infinity is futile, much like imagining the 4th dimension.
0/0 = ?
Infinity - Infinity = ?
Infinity/Infinity = ?
Infinity*0 = ?
I mention these because all of these relations are undefined. They can equal anything; the only time they have a precise value is when they are given context.
Take the equation X/X, for example. For any regular number, it is visibly equal to 1, whether that number is Pi, 42, 47, over 9000, 1.21 billion, or what have you. It keeps on being 1 no matter how big or how small X is, so we can infer that when X = 0 or X = Infinity, X/X still equals 1.
However, what happens when we have (e^X)/X ? As X goes to Infinity, e^X also goes to Infinity, so we have Infinity/Infinity again, but e^X approaches Infinity much faster than X does, so when X = Infinity, (e^X)/X also equals Infinity.
Infinity is like a regular number in many respects, but it's hard to work with in many ways.
Now, imagining a fourth spatial dimension...that warps your brain.
X = 0.999...etc.
X*10 = 9.999...
X + 9 = 9.999...
X*10 = X + 9
9*X = 9
X = 1
Think about it like this: the difference between 0.9 and 1 is 0.1, the difference between 0.99 and 1 is 0.01, and so on. What happens when you have an infinite number of nines after the decimal point? There's nothing you can add to it to make it 1. Can you have 0.00...(infinite number of zeroes)...1? You can try to make such a number, but it will be equal to 0, since all those infinite zeroes after the decimal point push that 1 into oblivion. So, in order to increase 0.999...etc. to 1, you must add zero to it, and if the difference between any two numbers is zero, they are perfectly equal. QED.
Infinity - Infinity = ?
Infinity/Infinity = ?
Infinity*0 = ?
I mention these because all of these relations are undefined. They can equal anything; the only time they have a precise value is when they are given context.
Take the equation X/X, for example. For any regular number, it is visibly equal to 1, whether that number is Pi, 42, 47, over 9000, 1.21 billion, or what have you. It keeps on being 1 no matter how big or how small X is, so we can infer that when X = 0 or X = Infinity, X/X still equals 1.
However, what happens when we have (e^X)/X ? As X goes to Infinity, e^X also goes to Infinity, so we have Infinity/Infinity again, but e^X approaches Infinity much faster than X does, so when X = Infinity, (e^X)/X also equals Infinity.
Infinity is like a regular number in many respects, but it's hard to work with in many ways.
Imagining the fourth dimension isn't that hard. It's just time, after all.Joshimodo said:
Now, imagining a fourth spatial dimension...that warps your brain.
In case you don't like the (rather elegant) proof that Lexodus provided, here's another.Lexodus said:
X = 0.999...etc.
X*10 = 9.999...
X + 9 = 9.999...
X*10 = X + 9
9*X = 9
X = 1
Think about it like this: the difference between 0.9 and 1 is 0.1, the difference between 0.99 and 1 is 0.01, and so on. What happens when you have an infinite number of nines after the decimal point? There's nothing you can add to it to make it 1. Can you have 0.00...(infinite number of zeroes)...1? You can try to make such a number, but it will be equal to 0, since all those infinite zeroes after the decimal point push that 1 into oblivion. So, in order to increase 0.999...etc. to 1, you must add zero to it, and if the difference between any two numbers is zero, they are perfectly equal. QED.
I had a discusion about this with a physicist friend and colleague of mine about when and if the universe would expand into it's self and if/when it does will this eventually become a quantium singularity, needless to say at the end both of us were more confused than when we started. Theres just some things we are not ment to know.Sheepzor said:
Well, It didn't really sound like a joke to me; actually if you mentioned the hilbert spaces then I would have copped, and it would have been quite clever. You sounded dismissive more than anything, and I hate people who have an air of superiority in everything they say. I know you know a lot about maths, actually giving your maths in the first place, and not 2 general, dismissive sentences would have been a lot better way to go about a good discussion.Maze1125 said:
Yes, I know I was generalising the definition of infinity, when infinity itself is rather poorly defined, and there are several different 'types' of infinity, but I didn't really want to into that. I was going to say "1/0" as an example, I just said any number so people less mathematically inclined would see the equivalence better: a lot of people don't think of 0 when they say number anyway. Of course, under scrutiny it doesn't hold at all, but if I go down that road I'd be a while.
Tell you what, I'll be more rigorous in my mathematical descriptions if you be less of a dick?
(Now that was a joke by the way!
)
no u need 0, 0s to get to 1. 0 is nothing therfor u cant have more than 1 of nothingRetodon8 said:
no 42 is the meaning of life watch hichhykers guide to the galix plzDaffy F said:
To tell you what infinity is imagine the Earth is a giant ball of steel and every million years and eagle lands on it and scrathes the surface. Imagine that and the time it takes to ear the ball of steel away to nothing. If can picture that you still haven't even touched on infinity that is how vast it is.
I don't at all follow you here, sorry.tanjiro6288 said:
I basically said that to reach 1, you'd need to collect an infinite amount of 0s, since you'll never get enough to actually reach 1.
If you could collect an amount of 0.000001s at least you'd reach 1 eventually, but that'll never happen with exactly 0.
Feel free to disprove that, but I think that both makes sense and is correct.
I think you're saying you need 0 amount of 0s to reach 1.
If 0s are the only thing you have (and 0 and infinity is what we're talking about here), then that's all there is, meaning you do need some amount of 0s... an infinite amount.
Not using any 0s is basically giving up before you even begin.
As for the second part... 0 is nothing, so you can't have more than nothing?
I don't follow that logic.
Usually a conclusion says something about the first part, the word before the "is", not the word behind it.
Compare to: A banana is yellow, therefore yellow (insert conclusion about yellow here).
If for simplicity's sake we say 0 and "nothing" are exactly the same thing, we could turn them around.
"Nothing" is 0, therefore you can't have more than 1 of nothing.
That's a statement, not proof.
I happen to have a whole bunch of nothing right in front of me this very instant.
(It doesn't take up any space, time, or anything, so it fits in quite comfortably inbetween the "things" that are.)
inf = inf + 1
inf^2 = (inf +1)^2
inf^2 = inf^2 + 2inf + 1
0 = 2inf + 1
-1 = 2inf
inf = -1/2
Huzzah!!!
inf^2 = (inf +1)^2
inf^2 = inf^2 + 2inf + 1
0 = 2inf + 1
-1 = 2inf
inf = -1/2
Huzzah!!!
Infinity isn't a number in the same way one is. Infinity is a concept; it is a term to describe a thing/system that continues forever. the best way to describe infinity as a concept is to check out a thought experiment called Hilbert's Hotel see attached link.
http://www.logicalparadoxes.info/hilberts-hotel/
http://www.logicalparadoxes.info/hilberts-hotel/
About your 0.99... being equal to 1. It is not. Your "proof" is ridiculous. It reminds me of the kind of thing people I used to tutor in remedial algebra would try. I thought you were joking at first, but realized you were either trolling or serious when you continually defended it.Maze1125 said:
Your statement that decimals are as exact as their ratio counterparts is misguided. If a ratio has a terminal decimal equivalent, then they are equal. If not, then our repeating decimal representations are not equal to the original ratio. They are simply the best representation we have in decimal.
Here's a tip: If you have to define a real number, the value of which is already known, your logic is flawed. So, the fact that you had to "define" 0.99... as "The decimal representation of 1/3, times 3," in order for your "proof" to work should give you a clue that any math professor in this country would laugh in your face and maybe kick you out of their class if you tried to make that assertion more than once. Here's another "definition of 0.99..." for you: 0.99 repeating is the greatest real number that is less than one.
Take a computer algebra class, and one of the first things they'll talk about is rounding and calculation error, and this is one of the calculations they'll talk about. Mathematicians and scientists all know about this, and that's one of the reasons they've come up with two solutions to the same problem. Math people hate it when you reduce ratios to decimal before the proof is concluded, and science people hate it when you mess up on significant digits.
The end.
What?PhiMed said:
I haven't actaully given a proof so far.
Here's one though:
An infinite decimal is defined to be:
lim(as n->infinity)sum(from k=1 to n) (a_k * 1/10^k)
where a_k is the kth digit of the decimal.
Therefore, 0.999... is defined to be:
lim(as n->infinity)sum(from k=1 to n) (9 * 1/10^k)
So all we need to do is show that that is equal to one.
Which is true iff for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10^k)| < e
Now sum(from k=1 to n) (9 * 1/10^k) is a finite sum, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10^k)| = |1/10^n|
So we need to show that for all e>0 there exists an N such that for all n>N |1/10^n| =1 then |1/10^n| e>0, then let N = 1/e and then |1/10^n| N
Hence the claim that, for all e>0 there exists an N such that for all n>N |1 - sum(from k=1 to n) (9 * 1/10^k)| < e, is true.
So, by the definition of a limit, lim(as n->infinity)sum(from k=1 to n) (9 * 1/10^k) = 1
Therefore, by the definition of infinite decimals, 0.999... = 1
QED
You are wrong.
0.333... is exactly equal to 1/3.
I can give a proof if you insist.
No, the exact opposite is true.
If you try to make an argument without clear definitions of what you're talking about then logic is impossible.
I never did that.
I never actually defined 0.999... at all before this post was made.
A lot of arguments only occur because people are using two different definitions of the same word without realising.
Wow, the irony of those two statements following each other is incredible.
There isn't a number that has the property of being the greatest number less than 1.
And I doubt a maths professor would kick you out of his class for that mistake, but he would certainly try and explain to you how the Real Numbers work.
If you've had a computer teacher who's claimed that 0.999... = 1 is due to a rounding error, then that teacher was wrong, and you should probably take your own advice and talk to a mathematics professor about it.
Yes, they do, because it's far less neat on the page, not because it's inaccurate.