Poll: 0.999... = 1
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We don't. They are two different representations of the same value. Only one of them is canonical. The other isn't used as much, as it is a whole lot more work that only adds to confusion. In fact, the notations of the form '0.(9)' or '0.999...' are often expressly forbidden, as they only add clutter as opposed to the canon form.Rainforce said:
There is never a difference because the amount of decimals is unlimited. The only difference is lexicographical and that does not affect the value or, indeed, anything. It's just so much easier to write the value's canon form, 1.Piflik said:
At least you finally admit that both notations represent the exact same value.
For those of you it's baffling, consider it like this. For every 9 you add, you're that much closer to one.
Ie,
.9 < .99 < .999
and so on.
So if the 9's on forever, you are drawing infinitely closer to 1. The more 9's you add, the more that distance closes. But the nines will keep adding forever, and so that distance will be forever reducing. The difference between 1 and .9999... will keep getting smaller to infinity. There is no value smaller than that gap, because it is infinitesimally small -- and because there are no infinitesimally small numbers that aren't zero . . .
Or try this on for size:
If you accept that 1/3 = .3333.... then you have to accept that .9999... = 3/3 = 1
Ie,
.9 < .99 < .999
and so on.
So if the 9's on forever, you are drawing infinitely closer to 1. The more 9's you add, the more that distance closes. But the nines will keep adding forever, and so that distance will be forever reducing. The difference between 1 and .9999... will keep getting smaller to infinity. There is no value smaller than that gap, because it is infinitesimally small -- and because there are no infinitesimally small numbers that aren't zero . . .
Or try this on for size:
If you accept that 1/3 = .3333.... then you have to accept that .9999... = 3/3 = 1
On the contrary, it is well accepted that 0.333... = 1/3, and you will really have a hard time arguing that one. The point I am trying to make is that 0.333... is defined as meaning 1/3, because that is what the mathematical term, visibly written on the page, means.Piflik said:
If you take 1 and split it into three, using division, you must by definition have 0.333...
You can't just have 0.3, 0.3, and 0.3. It doesn't add up to 1. So what's missing? Well, an infinite division effectively - that leftover 0.1 gets divided into 0.03, 0.03, 0.03, and there's 0.01 left over, and so on.
It works. Basic addition & division are founded on these principles. You need a recurring concept. That is infinity - that which recurs forever.
That you can't write it out is a limitation of language or the chosen form of expression, not the fundamental basis. That we have invented this handy ... symbol is a recognition and workaround of that limitation. It doesn't invalidate the concept at all.
Yes, all of maths is based on certain axioms, none of which can be logically derived, and so all of maths could be inconsistent with another kind of maths that used different axioms.Rainforce said:
However, the axioms mathematics uses are generally accepted to be self-evident and so simple they require no proof.
Funny how many people try to talk about what goes after infinity here. Nothing goes after infinity, ever. The difference between 1 and 0.9 repeating is nothing. If you actually try doing the subtraction yourself it doesn't take long to figure out that there will never be anything after the decimal point that isn't a zero. The nines go on forever, and you will never be able to subtract 9 from zero, so you'll always have to bring the one over from the left, and ten minus nine will always be one. It will always work that way. Zero minus nine doesn't ever equal zero, and you will never reach the end of the nines, so it just keeps going forever. Forever isn't really that hard of a concept to grasp. It just keeps going. There is no after forever. It's infinite. Infinities can have different ratios, for instance 0.3 repeating is one third of 0.9 repeating, or the limit of f(x)=1/x as x approaches zero is twice the limit of f(x)=1/2x as x approaches zero, but they won't end in different places because they don't end. Ever. That's what makes them infinite. The difference between 1 and 0.9 repeating is 0.0 repeating. The zeros go on forever. You can't say the zeros go on forever and then there's a one, because the very idea of "forever, and then" is wrong. They just go on forever. Nothing happens after that because there is no after that. (The difference between the limit of f(x)=1/x as x approaches zero and the limit of f(x)=1/2x as x approaches zero is actually infinity, by the way. Dividing infinities can produce odd results, subtracting them always produces infinity or zero. (Or technically negative infinity if you do it the other way round, but you get my point. It'll never be four.))
Infinity doesn't end. Please, if you're going to say something about what happens after infinity, do yourself a favor and shut your pie hole before you embarrass yourself.
Infinity doesn't end. Please, if you're going to say something about what happens after infinity, do yourself a favor and shut your pie hole before you embarrass yourself.
Yes...it is accepted, but that doesn't mean it is true...grammarye said:
0.3 =/= 1/3 correct?
let's continue...
0.33 =/= 1/3
0.333 =/= 1/3
0.3333 =/= 1/3
[.....]
0.3333.... =/= 1/3
There is still a difference between any decimal representation and 1/3, that is why it needs infinity to be represented and that's the reason why it is shady. Anything that needs infinity to be represented cannot be right.
Ok, one more proof to those still not convinced:
using the a_n = 0.9(n times) notation (a_1 = 0.9, a_2 = 0.99, ...) we have that:
1 - a_n = 1/(10^n).
1 - a_1 = 1/(10^1) = 0.1
1 - a_2 = 1/(10^2) = 0.01
so on and so forth.
so when n->infinity we have:
1 - 0.999... = limit of 1/(10^n) with n->infinity.
And as anyone who took basic calculus can tell you that limit is 0.
therefore
1 - 0.999... = 0
Seriously, it's the same number, different notations... And anyone who tells you different has no business hanging around math.
using the a_n = 0.9(n times) notation (a_1 = 0.9, a_2 = 0.99, ...) we have that:
1 - a_n = 1/(10^n).
1 - a_1 = 1/(10^1) = 0.1
1 - a_2 = 1/(10^2) = 0.01
so on and so forth.
so when n->infinity we have:
1 - 0.999... = limit of 1/(10^n) with n->infinity.
And as anyone who took basic calculus can tell you that limit is 0.
therefore
1 - 0.999... = 0
Seriously, it's the same number, different notations... And anyone who tells you different has no business hanging around math.
Not just not in a computer buffer. Actually NOWHERE. It is as you say purely a ruleset for manipulating symbols - there is no representation of it anywhere at all. And with this i do have a problem - there is a big difference between how normal people APPLY infinity, and what you're doing in maths - same words perhaps, but very different meaning.grammarye said:
And you know? Ever since i stopped being a science-fanboy by analyzing the theories i previously so blindly supported, and noticing one conceptual error after another.... i do not take unobservable unprovable axioms for granted anymore. Especially not when those very axioms are used to claim the biggest idiocies in modern science. It's one thing to create an axiomatic rule - its another thing to stop being aware that it is just that, and then infecting others aggressively with such falacies, via appeals to authority (when in fact, "the authority" doesn't understand its own basics).
No you don't. Select any two Real numbers, infinite length or otherwise. There will be an uncountable infinity of Real numbers between them. In fact, you can make a bijection of the set of Real numbers to any continuous open subset of the Real numbers, like (-inf, inf) (0.9999, 0.99991).Piflik said:
The only infinite-length numbers that look like they need special treatment are of the form x.y(9), as there are no numbers between x.y(9) and x.[y+1].
x.y(9) = x.[y+1]
For example, 0.(9) = 1
For example, 0.(9) = 1
No, no there isn't. You are superimposing your viewpoint of a decimal (that it must at some point finish) over the concept that this is a decimal that never finishes and thus cannot ever be precisely defined as a decimal. It's infinite, much as the Universe is and I don't see the Universe ceasing to exist because it's 'not right'.Piflik said:
I get the struggle - really I do - the human mind does not like to encompass concepts that are without limit - but the case remains that as you divide 1 by 3, you will never ever ever reach a point where it is 'finished' and the same holds true for the reverse operation. It must. Where exactly is the leftover bit going to go? Where is it going to hang around? Has 1 mysteriously lost a bit because the mathematical knife cutting the 1 cake has got a bit of sticky edge on it (I hate it when people do that)?
Wrong. Thats the thing about the Real Number system. You can't differentiate because the numbers aren't different in that system and you never need to.Piflik said:
Allow me to explain: Pi is an example of a number that has an infinite amount of decimals (that aren't repeating btw. The sequence is always unique). But Pi can still be measured and expressed, and formulas exist to calculate Pi, even though it's an infinitely long number.
There doesn't exist any way, however, that in the Real Number system allows you to express or formulate the difference between 0.999... and 1 (for the very simple reason that the difference doesn't exist).
Of all the mathematical proofs shown in this thread, someone posted this one which is the most effective counter to your post:
If the two numbers 0.999... and 1 are different, then you should be able to find the average of the two numbers with the following formula.
Average(A,B) = (A+B)/2
In this case: (0.999... + 1) / 2.
Problem is that you can't find an average between two numbers if they represent the same value. The average of 2 and 2 = 2 and the average of 0.999... and 1 is 0.999... and 1.
Here's the key to the whole thread:
Some people want to define infinity as "infinity plus rounding at the end towards an arbitrarily choosen reference" (how does the number know? Must be the mathematician)
Other people instead think, that infinity means just infinity, and that if one wants to do something on top of it, one needs to do something on top of it.
Some people want to define infinity as "infinity plus rounding at the end towards an arbitrarily choosen reference" (how does the number know? Must be the mathematician)
Other people instead think, that infinity means just infinity, and that if one wants to do something on top of it, one needs to do something on top of it.