1=.999999...?

[uppi]

No it is not objectively true. 0.999... is an element of (0,1), while 1 is not. Two equal numbers cannot be in a set and it's compliment. This is the real numbers.

If you want this to be a discussion, and if you want people to take you seriously, you cannot just ignore arguments. I have proven that 0.99... cannot be a member of (0,1). If you just keep repeating that without addressing that argument, you are showing the others that you are to be ignored in a discussion.

If I repeat something as much as I want, 10 times, 1000 times, 1000000000000000000000000000000000000000000000000000000000000000000000000000 times... and it doesn't achieve what I want to achieve - and if moreover I know for a fact that no matter how much countable times I try it won't work - how does it make sense to propose that doing so infinitely achieves it?
Can you answer this question? Infintely saying "I shall be a god" will not make me a god, because saying "I shall be a god" is well established to not make me a god. Similarly, if I add 0.09 to 0.9, it wont be one, and if I add 0.009 to 0.99 it wont me one and I can continue this until I grow old and die and it still won't be one and I know that my ancestors can do the same thing forever and it never will be one but simply stating that it goes on for forever makes it one because... reasons. Because it is consistent within mathematical rules. But the question how this illogical feat is accomplished never is even worth mentioning. Because math says it is true. Because a tautology says so.
Tautologies are by definition meaningless beyond their own system of axioms and conclusions... So IMO only a fool would even consider answering such an inherent contradiction with math rules. You may just as well say because Jesus told me.

You are right that math is meaningless beyond its own system of axioms and conclusions. But by asking the question whether 0.99... = 1, you imply that you are already within that set of axioms. You are questioning the validity of an infinitely ongoing process and you certainly have a point there, because nothing in reality is actually infinite. However, to define 0.99... you need an infinitely ongoing process. If you say that I can never sum all terms, you also have to say that I can never write down all the digits. If you doubt the existence of the sum, it would only be consequent to doubt the existence of the object itself. By using 0.99... you have signalized that you have accepted an infinite process and I consider it hypocritical to not accept an answer that also involves an infinite process.

In other words: If you ask a math question, you get a math answer. If it is not supposed to be a math question, we can discuss about that. But to have a meaningful discussion you first would have to define the meaning of the terms '0.99...', '=', and '1'. If I cannot presuppose the math definition of those, I have no idea what you mean. So if you want to continue this discussion, please define those terms.

 

[Tahuti]

If you think about it, maths are the only thing that can bring conclusive truths. A solid mathematical proof is not subject to interpretation and can not be overriden. Humanities are always a matter of debate and scientific proof are always provisional, meaning that those can be overriden by new scientific proofs using new instruments.

 

[Akka]

If you want this to be a discussion, and if you want people to take you seriously

Honestly, after beating a dead horse for pages and displaying arrogance and ignorance in the face of actual mathematic teachers by telling them they're wrong and he's right (despite actual academic evidences of the contrary), it's a bit too late to take him seriously.

This thread is just a Creation vs Evolution on steroids (because, as stupid as Creationists are, at least it's about the physical world and so there is not a strictly absolute answer, while mathematics are all about absolutes).

I'm surprised that nobody has found a way to claim that if you divide by 0 you get 0, and that he knows better than all these arrogant mathematicians.

 

[Kyriakos]

Honestly, after beating a dead horse for pages and displaying arrogance and ignorance in the face of actual mathematic teachers by telling them they're wrong and he's right (despite actual academic evidences of the contrary), it's a bit too late to take him seriously.

This thread is just a Creation vs Evolution on steroids (because, as stupid as Creationists are, at least it's about the physical world and so there is not a strictly absolute answer, while mathematics are all about absolutes).

I'm surprised that nobody has found a way to claim that if you divide by 0 you get 0, and that he knows better than all these arrogant mathematicians.

Actually if there would be any theorised 'outcome' possible for division by zero the end result would likely not be zero at all. It would be running towards +- infinity for any numerator that is real.

But <arrogance glasses> you are not a math teacher, are you?

 

[Akka]

Actually if there would be any theorised 'outcome' possible for division by zero the end result would likely not be zero at all. It would be running towards +- infinity for any numerator that is real.

But <arrogance glasses> you are not a math teacher, are you?

...

It seems that the entire point of this ironical example has flown a few hundred of miles above your head. I advise you to reread the sentence with its context.

 

[Kyriakos]

...

It seems that the entire point of this ironical example has flown a few hundred of miles above your head. I advise you to reread the sentence with its context.

Why should i, it is obvious you are way too smart for me

Don't be so moody.

 

[Borachio]

If you think about it, maths are the only thing that can bring conclusive truths. A solid mathematical proof is not subject to interpretation and can not be overriden. Humanities are always a matter of debate and scientific proof are always provisional, meaning that those can be overriden by new scientific proofs using new instruments.

What happens if I don't think about it? Are there then other things that can bring conclusive truths?

/silly

 

[onejayhawk]

If you want this to be a discussion, and if you want people to take you seriously, you cannot just ignore arguments. I have proven that 0.99... cannot be a member of (0,1).

I checked your posts for the last several pages. If you ever gave a proof, it is not in that frame.

Please reiterate, because what you are stating appears to be nonsensical. Only the limit of the series is not in the set. Since we are agreed the that stating that 0.999... = lim .9+.09+.009+... is equivalent to stating that 0.9999... = 1, please make no reference to that limit. It would be cheating.

You did say this:

So far nobody has addressed my proof that 0.99... is not an element of (0,1). The statement 0.99... < 1 is a contradiction to that and thus cannot be true. That proof involves no limits at all. So your position is not valid in standard maths, because it leads to contradictions.

This acknowledges a contradiction, which which seems to disapprove your assertion that 0.999 = 1.

I found it. It was a long way back.

So let us assume that 0.99... is an element of (0,1).
There is no number x that fulfills: 0.99... < x < 1 (I could formalize this step further if you do not believe that). Therefore 0.99... would be the maximum of (0,1), which is not possible by definition. Thus we have a contradiction, the premise has to be wrong, and 0.99... cannot be an element of (0,1). q.e.d.

You are still making the fundamental error, that a series has a limit. It has not. If a series is convergent, it is exactly one real number. It is the limit of its sequence of partial sums, but that is a different statement.

I will quibble with the definitions. A segment is open if it has no defined endpoint. You also resort to the same sort of squeezing that is tied up in the definition of limits. This is equivalent to the nonsensical "An open segment has no end point." Of course it does, you simply cannot find it. If it were true it would be a line (or a ray), not a segment.

Your statement that 0.999 is not < 1 is easily disproven. &#8721; 9 * 10^-n < 1, for all n > 0. QED

As you say in the quote above, this contradicts that 0.999... = 1. Hence, they are not equal. QED

It is not constructed any more than pi is constructed. It's a number with a value, and the value does not fall within the set (0,1) in the set of real numbers.

It is constructed. Here is the definition: &#8721; 9 * 10^-n, for n = 1 to infinite. As noted above, this is strictly less than 1.

J

 

[azzaman333]

It is constructed. Here is the definition: &#8721; 9 * 10^-n, for n = 1 to infinite. As noted above, this is strictly less than 1.

Except you cannot construct it because it goes on forever. If you try you will always end up less than 1 simply because you're physically unable to reach infinity, but then you're not constructing it you're constructing an approximation. So your process to define what the number equals (or doesn't, as the case may be) is flawed.

 

[Borachio]

Here is the definition: &#8721; 9 * 10^-n, for n = 1 to infinite.

Does that qualify as a definition? Isn't it just a way of writing it?

Hmm. Dunno. Maybe that's what definitions are, in the end.

But it looks to me exactly the same as 0.99... just with more notation. And, of course, there's no reason why my nines couldn't be followed by other numbers.

And having argued myself into a cocked hat, I'll be off.

 

[Leifmk]

Your statement that 0.999 is not < 1 is easily disproven. &#8721; 9 * 10^-n < 1, for all n > 0. QED

Nope. This only holds for finite n. The whole concept of an infinite series is being missed here.

 

[Kyriakos]

Nope. This only holds for finite n. The whole concept of an infinite series is being missed here.

Not sure how you can claim he misses the concept of an infinite series. He is just telling you (again) that while the limit of 0.9+0.09+... is 1, the ongoing and endless procedure of adding more parts of this sequence is not 1 but something having a limit to 1.

Can't have it both ways. Let alone that a sequence does not have to be regarded as equal to a specific fraction, so without arbitrary axiom-setting there hasn't been any reason stated by stuff like 1/3 can't be just an endless 0.3333... etc, and 0.99999... also is never reaching 1, which in a series constructed to go to a limit of 1 it also would never reach.

I love the 'this is first year uni math/calculus', though. Makes one think that in your and Atticus' view a uni math graduate is at least Gauss or something to us mere mortals. Knowing why something was set as something is the only thing that matters in the end, if you plan to construct more out of it and generalise, which surely even non-uni math degree holders know that math is about

 

[onejayhawk]

Not sure how you can claim he misses the concept of an infinite series. He is just telling you (again) that while the limit of 0.9+0.09+... is 1, the ongoing and endless procedure of adding more parts of this sequence is not 1 but something having a limit to 1.

Well put.

We live in a continuous world (I think), but we cannot grasp continuity in a rigorous mathematical sense. Fortunately, the continuous real numbers have more tractable embedded discrete sets, such as the rational or algebraic numbers.

Going back to the definition of open segment, a better definition is that there is no defined discrete set end point. The quoted definition is the Calc 101 version, not the more general one.

J

 

[azzaman333]

Not sure how you can claim he misses the concept of an infinite series. He is just telling you (again) that while the limit of 0.9+0.09+... is 1, the ongoing and endless procedure of adding more parts of this sequence is not 1 but something having a limit to 1.

That right there is fundamentally misunderstanding the concept of an infinite series.

 

[El_Machinae]

So, to phrase the current debate. Some people are saying

1 - 0.999... = 0

And others are saying

1 - 0.999... = undefined

???

 

[brennan]

Some people are asking "is there any a'priori reason for assuming that 1-0.999...=0?" and the answer appears to be "maths defines it that way". Which looks to me like a 'no'. Those saying 'no' seem not to realise what implications this has.

 

[Akka]

Nope. This only holds for finite n. The whole concept of an infinite series is being missed here.

Well, the difficulty to understand what "infinity" really means is the reason why this thread is such a beaten dead horse on all the world's forums. And why there is still people who can't get it.

 

[Smote]

It's invalid because I am talking about whether 0.999... or pi are legitimately decimal numbers to which we can apply arithmetic processes like multiplication in exactly the same way we use the number 7. Not whether 4x is 4x - which is algebra not decimal arithmetic.

I see what you're saying here: It's not immediately clear that a*lim(f(x)) = lim(a*f(x)).
This is like saying it's not immediately clear that 5*.999... = 4.999...

However, if our function is sum(9*10^-n), n = 1,2,3,..., then it is clear from the distributive property that 5*(a+b) = 5*a+5*b. This means 5*(a+(b+c)) = 5*a+5*(b+c) = 5*a+(5*b+5*c). Thus the distributive property can extend to any amount of numbers being added together like this. So, 5*sum(9*10^-n) does actually equal sum(5*9*10^-n) = sum(45*10^-n).

This looks like 4.5+.45+.045+.0045+... = 4.9999999...

 

[Smote]

For everyone arguing that .9999... != 1, do you agree that the limit as n-> infinity of the sum of the sequence 9*10^-n =1 ? (n = 1,2,3,...) (This series is 0.9+0.09+0.009+...)

If so, what does .9999... represent, if not this limit?

 

[Smote]

@Uppi and Atticus:

What the non-math degree posters claim (at least myself, but i think the other three are likely not saying that different things) is that we do not see just why it apparently is axiom-set that it is fine to use multiplication and like actions on a "number" like 0.99999.... which exactly has no set ending decimal part. Isn't this a bit like saying that an immobile object and a runner are both 'running' and thus we can view them as functions of 'running'? 0.9999... is not a specific number, and if you define it as the limit of sequences such as 0.9+0.09+0.009+... then isn't the set limit this sequence converged to the number 1? (which is outside of the sequence anyway?).

That is what we are asking (or at least mainly) : why is a limit taken to mean both the actual limit (1 in the above case) and an ongoing and non-ending reach for that limit? (0.999..... in that case). If you just set it axiomatically as that then we can accept it as an axiom, but axioms by definition are not provable within the context they define.

(Now, not from posts written, but from stuff inferred, i suppose there is some elaboration of this reasoning in current math, but we were not told what to look for at any rate)

(re Euklid, well, do not worry, i at least know what axioms are, let alone that officially they begin almost 4 aeons prior to Euklid, with Thales ).

https://www.calvin.edu/~rpruim/courses/m361/F03/overheads/real-axioms-print-pp4.pdf

See distributive property (axiom 3 in "operation axioms.")

Let's go through the logic. A number 4.556 = 4 + 5*10^-1+5*10^-2+6*10^-3.
5*4.556 = 5*(4 + 5*10^-1+5*10^-2+6*10^-3).

Now what are these numbers 10^-1? Multiplicative inverses. See Axiom 7 under Identity and Inverse Axioms. (For all x != 0, there exists y such that x*y = 1.) So 10^-1 is the multiplicative inverse of 10^1. 10^-2 is the multiplicative inverse of 10^2.

So, we have the sum of products of real numbers on the inside of the parenthesis, and it's clear that we can use the distributive property to multiply this by an additional real number.

.9999... is the infinite limit of the sum of 9* a bunch of multiplicative inverses of powers of 10. The distributive property applies to the series that we are finding the limit of, so we could multiply this series by another number and find the limit of that if we so chose. That is what 5*.9999... would signify.