A = .9999...., B = 1, for any arbitrarily small N, A + N > B. that means B can't be greater than A.

Y2k. Those aren't my words.

/mute

why would i assert something i can't prove? you're sitting here ignoring proofs for why .999... = 1 because it just doesn't make sense when you visualize it. no one is asserting things randomly, the fact that you think they are is telling. i'm done.

I know about limits: the concept is that if it was continuous it should be that number. For example, in the equation (x+5)/x as x approaches 0 the limit would be infinity, right? However, it would never actually reachInf out, right? Also, if you were to do the classic adding 1/2+1/4+1/8 it would never actually reach 1, correct? (If it did it would have to stop afterwards which, unless it reached it on a last digit, it would not and would become bigger than 1.) it is the same concept.

What if you add 1/inf, the definition of infinitesimal, to it? Arbitrarily close means that there will be an extremely small difference between them, it does not however, mean that the difference will be 0, which is required for them to be equal. I am going to bed now, so I may respond in the morning.

1 is always > 0.9999.... but for all practical purposes it may as well be 1. If you have 5000 FPS in a game it's still more FPS but it doesn't really do any more for you than 500 FPS so what's the practical difference?

If you write an infinite numbers letter x.... and you do this exactly 0 times... you get nothing cause you never wrote x even once.

@shield
what do you mean 1 is always > .99999...? It either is or it isn't. .99999... isn't a variable that is approaching a number. It is just a number. It's like how 1 isn't approaching 1.01 because the number isn't moving or changing. It also isn't approaching 1 because 1 is just 1. it's there. .999999... as a constant is just .99999... it's just there. From a logical standpoint, think of it this way:

if the last digit is 9, then that number isn't .9999.... because it has no last digit, it keeps going. So no matter how much you try to say that .9999...(with 10 billion 9s) is less than 1, you are true but that number isn't .9999... because the zeros are without end.

Also to everyone else, you CANNOT use basic math on hyperreal numbers. you can't say that .999999... is 1/inf away from 1 because how do you subtract .99999... from 1? where did you even get that number from?

It's like having 1+infinity. It is still infinity and is exactly equivalent to the original infinity. the only mathematical system where addition and subtracting actually affects infinity is under the surreal numbers system (Which is what tvrocks is referring to, but is still wrong because regular numbers do not follow the same laws as surreal infinities do, so you can't apply it to .999999...) You would need the hyperreal or ordinal or even cardinal system of infinity to find the answer. under these rules, you can use calculus, infinite series, or sets to manipulate infinity.

for the number .999999... you can use the sequence {.9+.09+.009+.0009+...}
this is the geometric infinite series: {.9+.9(.01)^1 +.9(.01)^2+ .9(.01)^3+...}

use the formula and you get exactly 1.

Thanks, Valis for rekindling this "debate"! If only you could have known the passionate responses this "question" can evoke ;)

Tvrocks: I repeat what I said in the previous thread. Following any of the different axiomatic schemes for defining real numbers (Cauchy sequences, Dedekind cuts, infinite decimals), the strings "0.9999..." and "1"represent the same thing. This is not something up for debate; it's an elementary proof and has been around for almost a century now.

If you want to debate the usefulness of the real numbers defined this way, go ahead. But unless you come up with your own internally consistent set of axioms, this is a moot discussion.

2 more things: infinity is not "a really big integer that's bigger than anything you can think of", it's not an integer (or rational/real number) at all. There's no reason why infinity/infinity has to equal 1, or has to even be a sensible thing to write down. Second, when you work with rational or real numbers, terms like "odd", "even" and "divisible by 3" don't really apply. For example, is 0.5 divisible by 2? Is 3.141592654... divisible by 2?Strictly speaking, everything is divisible by everything else (except 0), so that's not a very useful concept.

@Valis2501, your belief is irrelevant. This is not a matter of an opinion. If your friend has shown you the mathematical proof why 0.999... = 1 and there is no logical flaw in this proof, then 0.999... = 1. Now it is your choice whether you want to believe that 1=1 or not, but it does not change the reality.

I used to teach physics introduction (a couple of hours session) for 12 year old kids. I have always taken two balls which have different weight and the same shape and asked the kids to vote which they will believe will fall faster. Obviously, they fell in exact the same speed and time, and I have always had the pleasure to say this line: "The ball doesn't care how you have voted. Physics is about studying how the world works, and not voting about how it works".

Same applies to mathematics, but in a stricter way.

x=0.9999999.....(means periodical) so 10x = 9.99999999.... so further 10x - x = 9 this divided by 9 means x=1 so 0.9999999999.....= 1
my opinion, friendly greetings from Vienna

those proofs don't matter to tvrocks because no matter how much time he spends writing out 0.999999...., it will always be less than 1, so obviously 0.999999.... < 1. No matter how many additions he runs for adding 1/2 + 1/4 + 1/8 + ..., it will always be less than 1, so obviously sum(n:1->inf)1/(2^n) < 1

tvrocks, maybe this one helps:

you agree that there's no "last nine" in .9999999(cont.) (meaning it goes on forever), right?

so 1 - 0.9999999(cont) would be 0.000000(cont), which has no "last zero", meaning it is zero.

I would like to start off by correcting you on a general concept in mathematics that it seems that some of you don’t understand: just because something is a limit of a numer sequence does not mean that it would actually be equal to that number sequence. Limits are usually used in order to find what the value of x would be at an asymptote, however, the asymptote is actually impossible to reach. It is the same situation. The fact that it is the limit of something does not mean that it actually is that thing. Now onto the arguments.

The reason that they "don't matter" is because they're not truths at all, and are flawed. The classic 1/3=.333... 1=.999 is flawed as it assumes that 1/3 actually is equal to .333... even though if you actually divided 1 by 3 and .9 repeating by 3 you would see that, though they both end up with a result of .333… 1/3 will yield a remainder that .999.../3 will not contain (which means that there is a distance between them).


the x=.999 10x-x thing is also flawed as whenever you multiply anything by 10 it will move the number to the left, and there will be 1 less number after the decimal point. For example, 3.3(10)-3.3 does not equal 30, it equals 29.7 because there is one less number after it. Before you start saying, "infinity minus 1 is still infinity" try setting it up in an equation (ex, inf-1=inf, subtract inf) and you will see that, unless the laws of math are not true, then it would prove that it would be true in that situation. Frankly speaking there are no logical/ mathematical/ tested reasons of why your assertion would be true so it would be a way better policy to go with an assertion that is literally universal.

@basvan: this is not exactly an original axiom, nor did i have to come up with it, however, i do have one for you; all laws in mathematics are universally true (by definition) and, if an equation does not fit a law, then it is not true. For example, i believe that the inverse property of multiplication and addition are true and universal. I also believe in the property that when you do the same thing to both sides that they will still be equal, though i especially believe in the part of that property that says/ implies that if you do not do the same thing to both sides they will not be equal. (Except in the case of multiplying something by 0. I also believe in the multiplication property of 0, and that, as infinity fits the definition of being a number (a word or symbol, or a combination of words or symbols, used in counting or in noting a total.) that it can be used in mathematics.

@yoyo: Why do you think that the universal laws do not apply, while other things that have LITERALLY NO FOUNDATION EXCEPT FLAWED CONCEPTUAL STATEMENTS is true? The idea that it shouldn't apply because 1+inf is only on the surreal number system is also very weak as infinity is on the surreal number system in the first place. The geometric infinite series is also flawed as it does not find the actual number, it finds the limit of the number. Even if the number goes on forever, there will never be a point at which it becomes close enough to become one as 1. That is impossible 2. That is not how math works and 3. if it did theoretically become close enough to actually be 1 then it would still have more things after it as it would not end after magically becoming 1. It may be easier to grasp if you represent it with the equation .999...=[.9(.1^0+.9(.1^1)+.9(.1^2)]. in case i have not answered this, the idea that.999... is 1-(1/inf) comes from the general agreement that the equation .9(1)+.9(.1)+.9(.01)+.9(.001) will become arbitrarily/ infinitesimally close to 1, and infinitesimally is the definition of 1/inf .

@shield: I disagree with the idea that it is close enough for all functional purposes. They are 2 very distinct things. If you have 2 computers and give one away how many do you have, 1 or .999999... computers? (I don't know how that would even work.) Also, what if I want a number that is exactly equal to 1 and not just arbitrarily/infinitesimally close?

@kasimax: if have heard that concept before. the difference is actually 1/∞. look up the definition of infinitesmal. (I will also provide another example for you.)
https://en.wikipedia.org/wiki/Infinitesimal
use wallis's theory.

@tvrocks
You are displaying a basic misunderstanding of mathematical principles.

You are writing:
"Limits are usually used in order to find what the value of x would be at an asymptote, however, the asymptote is actually impossible to reach."

This is simply wrong. The asymptote is reachable - in the infinity. This is a concept you need to grasp. When you grasp the meaning of asimptote and infinity, it will help you to understand why 0.999... = 1.

But, you don't even have to grasp infinity in order to understand that 0.999... = 1. People has already proven it to you, by a simple mathematical equation. Please go through it carefully. If you will find a fault in its logic, you will be able to prove that 0.999... != 1. If you don't find a fault in its logic, then 0.999... = 1:

A = 0.9999999......
10 * A=9.999999..... (Multiply by 10)
10 * A=9+A (Write 9.999... as 9 + 0.999...)
9A=9 (Detract A from both sides)
A=1 (Divide by 9)

@baskineli: I have found a fault in the logic. :) It also seems that you don't understand both the concept of an asymptote and the concept of infinity. an asymptote is defined as being unreachable and it is also impossible to reach infinity. for example, in the equation (x-5)/x by your logic the y int would be infinity, instead of infinity just being the limit.

I can't wait to read tvrocks' committee-approved dissertation on this topic.

I realized i did not make the x-5/x thing very clear. if you believe that it would be able to reach infinity and to eventually become 0 then the limx as x->0 would not actually exist as there would be a limit as x->0^+ and a lim as x->0^- and they would be 2 different things, and it would mean that it would actually have 2 y intercepts (inf and -inf) and would not be a function.

Functions do not have to be continuous.

If any of you believe that there should be different rules for infinity please 1. explain why the universal rules of mathematics would not apply, (saying infinity is not a number is not true as a number is definied as being a quantity or amount, and infinity can be used to express a quantity, so, by that defintion woudl be a number and can be used in math. If you do not believe it is a number please also try disproving the idea that it is an amount and/ or a quantity. That property of it alone should allow it to be used in math) 2. why those new rules would apply (please also provide other concrete examples of those rules) and 3. why those rules should be used to establish your point. Thank you. 1/inf=/=0 . inf/inf=1 inf+8>inf .

Infinity can't be quantified. Therefore 1 is wrong. Therefore you are wrong.

jeff, a function does not have to be continuous, however, it can only have 1 possible y output for any x input in the function. If at x=0 it equals both infinity and -infinity, as it would if the limit actually does means that it actually is that number, then it would not be a function.

tr, how many 9s are there after the decimal point in .9 repeating? infinite, right? isn't that a quantity?

A quantity can be counted. Infinity can't, by its very nature and definition.

here is a google definiton i found of quantity: "A specified or indefinite number or amount:" if it can be used to measure how much there is of something, or if it does have value then it is a quantity.

The function x-5/x is undefined at x = 0. It doesn't have values of infinity. That's one serious flaw with your logic.

EXACTLY! I used it to disprove the concept that the limit of a number is the value for that number at that value which seems to be a very common argument for .9 repeating being 1.

the concept is that it will become infinitesimally close to 0 without ever actually reaching it. It is the same situation, and that helps to disprove the idea that if it becomes infinitesimally close to something that it will be that number.

It's pretty close.