For there to be continuity, don't we need numbers to be infinitely close?
For there to be continuity, don't we need numbers to be infinitely close?
This all goes back to the concept of infinity.
Imagine that such a number 0.000...01 did exist.
There would be an infinite number of 0s, and therefore you never reach the one. The entire concept of a non-terminating decimal is that you cannot pick a digit and say "this is the last digit" because you will never get there. Every time you might reach the 1 at the end, there are still infinitely more 0s before you can reach it.
its really simple when you look at fractions
like 0.3˘ x 3 (dunno how to do right symbol)
What are people trying to prove here? Math, or what numbers represent and how they work?
If we take 0.99999 and instead of "reading it"from the first 9 we "read it" on the 9 in the middle, we can put a finite number of 9's on each side. If we keep adding 9's to the left and to the right of the middle 9, we'll have an infinite decimal with infinite 9's to the left of it. We can't identify how many digits you have to go to reach the first 9, it's infinite. That doesn't mean it's not there, if it weren't the chain would never exist.
What you're saying doesn't make sense. The very idea that the decimal extends infinitely makes it impossible to choose a "middle".
Edited:
It's people who know math vs. people who think they're going to disprove the work of educated Mathematicians who gave us the evidence we're arguing.
Middle can be any digit that is infinitely far away from the extremities of the number, the argument stays the same.
Let's take a semi-line if you prefer. It has a defined start but stretches to infinity. if you place the origin of your referential on infinity, the origin of the line will be located at infinity, but it will still be there, it'll exist. Not in a finite distance, but it'll be there.
0.999... = 1 because we don't allow for infinitely small numbers in our math.
This does not prove anything.
People have posted mathematically solid evidence for the past 5 pages. Arguing against this is null, the people who posted the evidence didn't make it up, they got it from websites where Mathematicians proved them.
I didn't say their evidence was faulty. You can come up with advanced mathematical ways of showing 0.999... = 1, but all they're really showing is that we don't allow for infinitely small numbers in math.
Edited:
Using infinity to prove something is on par with dividing by zero. As far as we know, a true zero or a true infinity does not exist in our universe, so ultimately we are going to make up its properties, so of course 0.999... = 1. That's what the made up properties of infinity tells us.
Every single digit is infinitely far away from the end of the number. 1 is the same distance from infinity as 1 million is. There is no "middle" to infinity, because if there would it would not be infinity.
0.111... = 1/9
0.222... = 2/9
0.333... = 3/9
0.444... = 4/9
0.555... = 5/9
0.666... = 6/9
0.777... = 7/9
0.888... = 8/9
0.999... = 9/9 = 1 end of thread
I feel like this thread should have ended at the OP because there was a perfectly reasonable proof right there. But some people are like "Fuck that; I may be a high-school student who failed algebra 1 but I know better than all of the mathematicians in the world!"
Wat.
What? No shit. I said extremitieS. I.E. Infinity and the first digit.
A digit that is infinitely far away from the extremities is infinity. So there is no middle.
ITT: People in thread do not understand concepts of higher mathematics such as, infinite series, induction and convergence.
Holy fuck
what are you
oh my god
What? The "middle" of an infinite series is in infinity itself. Different infinities, that's why 2n - n tends to infinity and not 0
I used to think this once.
Until I saw the proofs.
The fact that numbers are infinitely closely packed in the real line is exactly why you can't pick a number adjacent to another number.
Edited:
Definitions for continuity are carefully chosen so that they rely only on finite numbers.
Isn't this just proving that our ways of thinking are broken rather than making sense?
If you pour a glass of water into 3 glasses with exactly the same amount in the 3 glasses, it's 1 glass devided in 3. Saying it's 1/3 is fine, but saying it's 33, etc % is wrong. Just because we can't devide 1 into 3 efficiently doesn't mean it's content is 0,9. Assuming that this flaw makes it right to say 0,9 is actually also 1, is just as flawed or even worse.
Besides, 0,999 is infinitely far away from being 1. Just because infinity isn't a number doesn't mean it's not there. And even if it was a number, it can never become something between 0,999 and 1 unless you make up some new rules.
But yeah i am probably too dumb for this with my elementary level math.
watch as people argue that your logic is flawed and that you are stupid
Well, I don't mean picking a number just like that. I can't say how it looks like or write it, I can only say it's infinitely close to a number. but if that means the difference is 0, then it is the number itself. So doesn't it form an infinitesimal gap?
I am confused.
Never mind, it's a bad argument.
Euler is rotating so fast in his grave that I believe we could use it to power Switzerland for the next few decades.
Nobody here is talking about 0.999 but you. We're talking about 0.9...
If you pour a glass of water into 3 glasses with exactly the same amount in the 3 glasses it's 1 glass divided in 3. So there is 1/3 in each. 1/3 = 0.3... Multiply it by 3 and you get 0.9... which is everything you had, which is 1.
1 = 3/3 = 1/3+1/3+1/3 = 0.3...+0.3...+0.3... = 0.3... x 3 = 0.9... These are all equal.
duuuude its not 1 its smaller than one CANT YOU SEE THE FUCKING 9!
http://www.wolframalpha.com/input/?i=0.9...
Edited:
ahaha the link doesnt want to put the ellipses in so we are getting 9/10
I only watched this to fap to the sound of her voice.
My math teacher did so many proofs on this... Can we all just agree .999(Repeating) = 1.
There is no definable 'middle' to an infinite string of numbers, but your second point is interesting. An infinite line is, in every possible way a person might care about, equivalent to a circle, with the north pole representing the point at infinity. So we can define various properties at infinity for 'things' (functions, etc) acting on the real numbers by the way those functions act on circles at the north pole. But this isn't relevant.
No one's sitting around saying 'I won't allow for infinttesimals!'. Their non-existence is a direct consequence of our definition of the real numbers, which is what you happily use everyday without question.
Choose two distinct numbers a and b. The difference between them is (a-b)/2. This is non-zero since a and b are distinct. Moreover, the difference is definable as long as they are distinct--it may be very small, but it is definable no matter how close they are together--and 0 if and only if they are in fact the same number.
So what is the difference between 0.999... and 1?
nope because apparently people don't know what repeating decimals are, to begin with
Yes and you lose 0.1 making it impossible for it to be 1 again.
My brother told me another way to think about it.
What is 1 minus 0.9...
The answer is 0.0...
That is a fancy way of saying zero. So if 1-0.9... is 0 they must be the same value.
No, you're just going to have the idiots saying that it equals "0.0...1" which makes no sense.