Poll: 0.999... = 1

[emeraldrafael]

YES! Cause something's missing! What if I lost a toenail? You justs aid that they are different, its just thats tiny. Germs are tiny, do they not exist?

ummm... that's not what I meant. the difference between you and you losing a strand of hair is so small that it pretty much does not exist. you are (1 whole) - (1 strand of hair). The strand of hair is small enough to not make a difference. You are still 1 whole.

in the example you make, the germ is the 1 whole, therefore it (no matter how small) exists.

Alright. Then if i were missing a single cell in my body, would i still be the same? Maybe, maybe not, dependingon the cell and its importance. But the fact that I'm missing it is what makes me not what I was before it was gone.


So, my college professor, who has been teaching at the college for what he says is 40 years, but administration says he's been there 50 years. Either way, he's head of our math department and does a bunch of those lecture things since the guy's easily.. like... 70, if not 80, at other colleges. ANd here's how he explained it to me, which is in my words, not his, so dont get on my back about if I didnt phrase this right and he's dumb. But I will give it to you in the same way he gave to me, which was a numbered list becuase he now thinks I'm retarded since he said he covered this in college algebra and I wasnt listening (even though I clearly remember this).

1: Infinity is range, not a number. Its something to encompass everything, and to say something is never ending, since some smartasses like to add an extra zero to a number and say the made a new number. Infinity has no value, but is rather the combination of all values (which, i guess if you think about it still brings infinity to a value of zero since all numbers would cancel out their positive/negative forms).

2: HUmans, by nature, like things to have value, or physical existence. They dont like the concept of God because he's not tangible. They like numbers in the same way, which is why we dont like to play with imaginary numbers, because I cant say to you that Johny has one imaginary apple. YOu take that as johny pretends to have an apple, which means he has none and is insane to beleive otherwise.

3: Hence, humans can not be happy and satisfied with the number infinity as it is not a value, but range. You can not have infinity dollars physically in the same regard as you can have 1, 2, 5, 10, 100, a million dollars in your hand to hold. Those numbers are tangible, which is what humans like, as it helps them to relate.

4: .999...! =/= 1 in the sense that they have values. 1 > .999....! because it is not the whole one when you look at it on the plain of values in numbers (compared it to have .999...! % of an apple. Though it maybe tiny and insignificant, its still not the whole apple).

5: Numbers have values when pulled from infinity. Infinity in and of its self is another plain of mathematical reality, as it is range that does not top and is hard to tell where it even began, because infinity doesnt have a starting value.

6: Therefore, .999...! = 1 in the plain of range. However, much the same in the plain of infinity as a range, 1 is no different then 10 and is no different then -67. in the range of infinity, where nothing has range or value, -67=1=10 because there is no physical measurement. However, when you look at a number, you are giving it value. 1 is a value, you have one of something. 2 is a value, you have 2 of something (apples, again). if you place two physical apples down on one table, and one physical apple on the other, you have more apples on the first table because you can see and measure it. This is (what he calls) the human value plain of math, where numbers have value, a starting and an ending point.

7: Because you give a number value when you pull it from infinity, it does not equal another unless they are the same value, no matter how close the numbers are. .999...! =/= 1 on the human value plain because if you were to put to separate cups, one with each value, the one with 1 (liter, cup, gallon, pint, whatever) will have more and weigh more then the cup with .999...! since you have given it value.

On the infinity plain of math, you can have both equal, because nothing has value. The human mind doesnt have to look at it as two apples against one. or .999...!th of an apple against 1 whole apple. BUt when you work with numbers, you give them value. Even imaginary numbers have imaginary value, but because they are imaginary it makes it difficult to do anything with them the same way you do "real" numbers.

If you want to look at it another way, its like the difference between the plains of 2D and 3D. You have a line/rod (a 2 dimensional object that when talking about a line is on the infinite plain, while a rod is on the value plain, as it as beginning and ending), can not exist in a 3 dimensional world on a 3 dimensional plain because it misses a dimension. It appears flat to the human mind, and will always appear on a 2D plain. A brick, 2x4, even pencil, anything that is perfectly straight and level that you can physically hold is a 3 dimensional line. In much the same way, 3D can not exist on a 2D plain. You can not have a cube on a single piece of paper that pops out from a single drawing. All "3D" art/paintings are flat objects that appear 3D through optical illusion. However, if you try to grab it, its still flat. Its two separate plains, each with its own rules and meaning of existence/value/definition of what makes something what it is.

So no, .999...! can not equal 1 exactly unless you go up to the infinite plain where numbers lose value. TO ask it then, you have to say does a portion of infinity equal another portion of infinity, without giving a specific value to it. To say it in the way .999...!=1 is to mix two plains of reality that dont mix on simple basis that one has a separate definition of reality then the other.


SO yeah, thats what he said, thats what I'm going with. You're mixing separate plains. its like trying to mix 3D with 2D. Trying to make a 2D object pop out into 3D, and trying to make something 3D out of a flat 2D object.

 

[Soraryuu]

1: Numbers represent reality. Math is based on our perception of it. So, when we don't know something about reality, we can't know about the same thing in math. That's why it's relevant.

You seem to have a pretty strange idea of what constitutes reality, i.e. that a number doesn't exist unless you can string together that many particles. Does pi not exist then? Pi is irrational, so like .9999..., it never ends, but unlike .99999..., is also never repeats itself. So, by your logic, pi can't exist unless you can divide matter into infinitesimals! Congratulations, you've just disproven circles!

And please don't pull a "if I can't understand it it's not true" argument.

Pretty ironic coming from someone who, over the course of this thread, has been given at least five different mathematical proofs for why this is true and still refuses to accept it.

That works too. The last decimal it needs to keep it's 9 doesn't exist. There's your answer.

Uhm, no. Neither decimal ends, so you always have something to subtract.

Here's a question for you. If 1 and .999... are two different numbers, then there must be some other number between them. What is it?

1: Ah, I didn't think about numbers representing dimensions. Well, in that case, congratulations! You just proved infinitesmall numbers exist. That makes thing more clear.

2: I meant going "your argument is bullshit(because I don't understand it) so mine still stands" without giving counter-arguments. You don't, and neither have I(yet). Also, you really expect me to go through 18 pages just for something that small?

3: I've come to realise the original equation isn't correct. This is how it should be done:

y = 1 - (10 * n)
x = 1 - y
2x = 2 * (1 - y)
2 * (1 - y) - 1 * (1 - y ) = 1 * (1 - y)
1 =/= 1 * (1 - y)

Usually when 1 = something other than 1, there's just a bit of bad math going on.
(like me making typos in the equation)

EDIT: Just saw the post above me. Man, my home-made logic falters against that of someone who is past 9th grade.

 

[Athinira]

SO yeah, thats what he said, thats what I'm going with. You're mixing separate plains. its like trying to mix 3D with 2D. Trying to make a 2D object pop out into 3D, and trying to make something 3D out of a flat 2D object.

Which is why we define which number system we are working with.

If we are working with the Integer system, then decimals doesn't exist. There is only 0, 1, 2, 3 etc.

If we are working with the Natural Number system, then negative numbers doesn't exist.
If we are working with the Real Number System, decimals exist, but infinitesimals doesn't.
If we are working with the Extended Real Number Line, then infinitesimals exist (at least as an idea/concept).

Establishing what number system is being used is paramount, since it defines everything. Take a simple equation like 30 / 8. Most people will probably calculate on this and say "That's easy. It's 3.75". But if i established that we were using the Integer system, the correct answer would actually be either 3 or 4 (depending on whether or not you round or just strip the decimals), and not 3.75.

Humans typically work with the Real Number System, and in that system, 0.9r equals 1, because there cannot exist a value between the two. In the real number system 0.9r IS the whole 1.

 

[Nouw]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

Doesn't that imply 9x=10/x?

Nope. Where do you get the divide sign from? It shows that 9x = 9 so x = 1

Woops typo . Well at least I think I get it now.

 

[zoulza]

1: Ah, I didn't think about numbers representing dimensions. Well, in that case, congratulations! You just proved infinitesmall numbers exist. That makes thing more clear.

Oh dear. I really hope I didn't. The reals do not permit infinitesimals. And that, I think, is the problem people are having with this: they think that .99999... is smaller than 1 by some "infinitely small" amount, when no such amount exists.

3: I've come to realise the original equation isn't correct. This is how it should be done:

y = 1 - (10 * n)
x = 1 - y
2x = 2 * (1 - y)
2 * (1 - y) - 1 * (1 - y ) = 1 * (1 - y)
1 =/= 1 * (1 - y)

Usually when 1 = something other than 1, there's just a bit of bad math going on.
(like me making typos in the equation)

EDIT: Just saw the post above me. Man, my home-made logic falters against that of someone who is past 9th grade.

I'm not sure what you're trying to do with those equations, so let me show you a different proof for why this is true. The number .99999... can be written as an infinite sum of 9 * 10^-n (i.e. .9 + .09 + .009 + .0009 + ...). Take the limit of this, and it is 1. Absolutely, if you truncate it at some finite n, then the resulting value will be less than one, but you're not. You're sticking an infinite number of nines in there and each one you add brings you closer to the limit (which is one). The thing is, with .9999..., you're not adding any nines, they're all already there!

Perhaps you should try looking up the definition of a limit.

 

[Maze1125]

Snip

I don't believe a mathematics professor would have said any of that. So either you're making it up entirely, or you've completely twisted what he did say. The entirety of that was vague, imprecise and hand-waving. The exact antithesis of mathematics.

On the off chance he did actually make the explicit claim that 0.999... =/= 1, could you please show him the following proof and tell me what he says?

An infinite decimal is defined to be:
lim(as n->infinity)sum(from k=1 to n) (a[sub]k[/sub] * 1/10[sup]k[/sup])
where a[sub]k[/sub] 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[sup]k[/sup])
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[sup]k[/sup])| < e

Now sum(from k=1 to n) (9 * 1/10[sup]k[/sup]) is a finite sum, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10[sup]k[/sup])| = |1/10[sup]n[/sup]|

So we need to show that for all e>0 there exists an N such that for all n>N |1/10[sup]n[/sup]| =1 then |1/10[sup]n[/sup]| e>0, then let N = 1/e and then |1/10[sup]n[/sup]| 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[sup]k[/sup])| < e, is true.
So, by the definition of a limit, lim(as n->infinity)sum(from k=1 to n) (9 * 1/10[sup]k[/sup]) = 1
Therefore, by the definition of infinite decimals, 0.999... = 1

QED

 

[Lukeje]

Math is axiomatic and absolute. If a system or a theory says something works in a certain way, then it just does, within that system. There is no intuition, there is nothing to comprehend, there is only Math and its laws, as defined by the System's Postulates. If you deny an axiom and substitute your own, you create a new system with a new ruleset. If you do it as a part of a proof, the proof is invalid in the original system and therefore irrelevant.

Sufficiently advanced math is indistinguishable from magic.

That is part of my dislike for maths and how it is abused to for real-world modelling tasks, in which it doesn't belong. See my P.S. in my previous post. If weird rules just affect stuff inside a theoretical system - fine. If those rules result in massive logical breaks when making explanations about reality, not fine.

Can you explain where this leads to a logical break in our explanation of reality please?

How many infinities (including mathematical points) can you spot in physics? Why is it that the maths which use those do work, and yet, the infinities themselves have never been observed directly? Here's a hint: perhaps the most reasonable use of a mathematical point, for calculating reality, is like a "position-marker" (while never actually using it infinitely precise... its just there in the models, and then in practice gets enough precision as necessary).

P.S.: I could go into more detail, about how numbers and the ranges in-between, are directly derived from how we "address" things in our perception - and thus, actually come from something very intuitive and imaginative... and how this mechanic resulted in a lot of misunderstandings.... including the wave/particle dualism... but this would derail the thread too much.

Then this seems to stem from a misunderstanding of physics as well as of mathematics.

 

[Maze1125]

If we are working with the Extended Real Number Line, then infinitesimals exist (at least as an idea/concept).

No they don't. The Extended Real Numbers only add the values of infinity, either both positive and negative infinity, or just infinity. They don't add infinitesimals. You have to go as far as the Surreal Numbers to get infinitesimals, and even then I've yet to see anyone prove that 0.999... =/= 1 even in the Surreal Numbers.

Establishing what number system is being used is paramount, since it defines everything. Take a simple equation like 30 / 8. Most people will probably calculate on this and say "That's easy. It's 3.75". But if i established that we were using the Integer system, the correct answer would actually be either 3 or 4 (depending on whether or not you round or just strip the decimals), and not 3.75.

Not quite, if you're working in the integers, then 30/8 doesn't have a solution, you don't just round.

 

[CrystalShadow]

But isn't math begging the question by default?

Eh. Maybe not. But for all it's inherent logic, Math in the end still comes down to it's axioms. And those axioms are completely arbitrary.

They have to be, because if you can logically derive them, then you can decompose them into a combination of other parts that logically lead to them.

Therefore, the most fundamental axioms cannot themselves be logical statements, or you would have a infinite regression.

Fundamental problem with logic all round to be honest.

Logic, fundamentally, isn't logical. XD

Actually it isn't. You are confusing "logic" with "proof". It is true you can't prove logic with logic. The foundations of logic are things like a=b and a!=!a. Everything kind of built upon that. Those things are then used to prove things and define others. Although you do seem to be getting more philosophical than this thread would probably like.

Most likely, yes. I am getting a bit philosophical here. Unfortunately I have difficulties with abstract concepts to avoid doing that.

I guess the philosophical things I struggle with are only tangentially related to maths, but it essentially derives from people saying 'be logical' as if that is a meaningful distinction from whatever alternative there may be. (How can it be, if the origin of a logical statement cannot itself be logical?)

But... Not quite the right topic for a discussion like that.

 

[Rubashov]

Depends.

What degree of accuracy am I recording to?

If it's an integer, and I'm using the round function, then it's one. If it's not using round, then 0.9 = 0. Yay truncation!

Short? Long?
Single? Double?

Stop thinking like a computer scientist. We're recording to infinite accuracy. There is no truncation and there is no rounding. We're not claiming that 0.999... is approximately 1, we're claiming that it IS 1.

 

[Rubashov]

That doesn't make sense. You're saying that 10x-x is a number with an infinite number of decimal places occupied by nines, but the last decimal place is occupied by a one. Which means that you're essentially saying that 10x-x has both an infinite number of decimal places and a finite number of decimal places. That's a contradiction.

You cannot perform mathematical operations on an infinitely repeating number. Therefore, you must at some point terminate the string. At that point, you can then multiply it by 10 and proceed.

However once you do that, 9.9999...999 will have shifted to the left, so 0.999...999 will have one more significant digit. Thus, you get 8.999...991.

Edit:

But it goes on to infinity, so technically there's no 1.

Same answer to you too.

Of course you can perform mathematical operations on an infinitely repeating number. Here's an example: 3 times (1/3) = 1. You can also perform mathematical operations on irrational, and therefore non-terminating, numbers: sqrt(2)^2 = 2. Would you also deny that the square root of two times the square root of two is two?

 

[Sebenko]

Depends.

What degree of accuracy am I recording to?

If it's an integer, and I'm using the round function, then it's one. If it's not using round, then 0.9 = 0. Yay truncation!

Short? Long?
Single? Double?

Stop thinking like a computer scientist. We're recording to infinite accuracy. There is no truncation and there is no rounding. We're not claiming that 0.999... is approximately 1, we're claiming that it IS 1.

Uh... no.

I'm a computer scientist, it's what I do.

Do you really need infinite accuracy? I doubt it.

It's all very exciting as a hypothetical question, but I don't think there's ever been a real world situation where recording to infinite accuracy has been the only option.

 

[Rubashov]

So you agree with my prof that 0 = 1? Because if you want to do traditional maths with infinity, you would have to...

Infinity is an integral part of the so-called "traditional" math. Infinity and infinite numbers are also very prominent in the Set Theory, look it up.

You seem to have no understanding of how math actually works, so would you kindly post your alleged "proof" of 0 = 1?

I find you lack of math disturbing.

I did in the first post you quoted, but here it is again:

1-1 = 0
(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1) = 0
= 0
1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+..... = 0
1 = 0

Summation((-1)^n) from n = 0 to infinity does not converge; it's value is undefined. That is to say (1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+..... does not equal zero and 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+..... does not equal 1. Neither equals anything because they are both divergent.

Summation(9/(10^n)) from n = 1 to infinity does converge, however; in fact, it converges to 0.999... or, in more familiar notation, 1. So your example is irrelevant.

 

[Rubashov]

Depends.

What degree of accuracy am I recording to?

If it's an integer, and I'm using the round function, then it's one. If it's not using round, then 0.9 = 0. Yay truncation!

Short? Long?
Single? Double?

Stop thinking like a computer scientist. We're recording to infinite accuracy. There is no truncation and there is no rounding. We're not claiming that 0.999... is approximately 1, we're claiming that it IS 1.

Uh... no.

I'm a computer scientist, it's what I do.

Do you really need infinite accuracy? I doubt it.

It's all very exciting as a hypothetical question, but I don't think there's ever been a real world situation where recording to infinite accuracy has been the only option.

What's needed in the real world is irrelevant because the question at hand is a conceptual one. I'm a computer scientist as well. However, I recognize that the limitations that reality imposes on computers don't apply to questions about mathematical concepts. The mathematical concept 1/2 is never 0, even though the value a computer returns when integer 1 is divided by integer 2 is 0.

 

[Maze1125]

Depends.

What degree of accuracy am I recording to?

If it's an integer, and I'm using the round function, then it's one. If it's not using round, then 0.9 = 0. Yay truncation!

Short? Long?
Single? Double?

Stop thinking like a computer scientist. We're recording to infinite accuracy. There is no truncation and there is no rounding. We're not claiming that 0.999... is approximately 1, we're claiming that it IS 1.

Uh... no.

I'm a computer scientist, it's what I do.

Do you really need infinite accuracy? I doubt it.

It's all very exciting as a hypothetical question, but I don't think there's ever been a real world situation where recording to infinite accuracy has been the only option.

The issue of accuracy is irrelevant.

0.999... = 1 if you round and also if you don't. The statement is true both ways. So pick what ever accuracy you want.

 

[Soraryuu]

Ok, blank slates now. New angle of attack. So, 0.999_ is a number that has "infinite" nines? You're using infinity in a number?

Infinity is not a number.

Since it's not a number, you can't multiply with it.

Since you can't multiply with it, you can't count with it.

Therefore, any repeating number is false. Incomplete.

1/3 is impossible to make into a complete number, same for all other fractions that lead to repeating. All of this 0.999_ stuff is a mathematical mistake on par with 1/0.

In other words, bad math. Don't drag forth your equations before you disprove this.

 

[Rubashov]

Hey, let's try something.

Let n = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... and so on, forever, involving all negative integer powers of 2.

Now, 2n = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... and so on.

This is 1 more than n.

2n = n + 1

Subtract n...

n = 1

So what you're saying is that n is actually infinitesimally smaller than 1, huh?

Would that make it equal to 0.999..., since that is also infinitesimally smaller than 1?

But I've just shown it's one.

What am I doing wrong? Where is the flaw that shows that when n + 1 = 2n, n =/= 1?

Again your geometric series...I said it before and will say it again. A limit is not a value. It is a limit and will never reach that value.

The limit of the sum 1/(2^n) for n -> infinity is 1. That much is true, but the function will approach this value asymptotically and never reach it.

The value of an infinite series is defined as the limit of the sequence of partial sums. Even a cursory study of infinite series will show that this is true. I suggest you get on that. [http://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx]

 

[Sebenko]

The issue of accuracy is irrelevant.

0.999... = 1 if you round and also if you don't. The statement is true both ways. So pick what ever accuracy you want.

If it's 1 in any case, why is it being discussed?

 

[Optimystic]

Therefore, 10x - x = 9

I sense a disturbance in the Force...

 

[emeraldrafael]

Snip

Of course you're not going to believe me, cause its different then whats being siad. BUt thats what he said. You're working on two different plains of mathematical reality. Seems pretty straight forward to me. When you give a number a value, you cant have it go to infinity doesnt have value. Besides, the guy's been teaching and doing math for more then 50 years of his life, I'm pretty sure he knows what he's saying. How do you argue what he says? What is a value of infinity on a value system?

Snip.

Yes, but those all have values. Thats what he's saying. He's saying that as long as the number is real and has value, it cant equal another without rounding. So whether its an interger, natural, whatever, it has value, so that human mind can comprehend. He's not saying that .999...! cant exist, just that when you give it value, like that because its a written out number, it can not be 1 because one as a value is more then .999...! will be.