# 0.999... = 1 debate

Discussion in 'Math' started by BestFriend, Oct 16, 2010.

1. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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Let this be the thread discussing the proof of 0.999... = 1

This is their first argument and proof:
(1/9) = (0.1111...)
9(1/9) = 9(0.1111...)
1 = 0.9999...

is similar to this argument
(1/4) = (0.2500...) = (0.2...)
4(1/4) = 4(0.2...)
1 = 0.8...

Which is erroneous.

Second proof: Digit Manipulation
(x) = (0.9999...)
10(x) = 10(0.9999...)
(10x) = 9.999...
(10x)-x = (9.999...) - (0.9999...)
9x = [9] [9 is ERRONEOUS, the answer is 8.9999... look at the argument below]
x = 1

is similar to this argument:
(x) = (0.9999)
10(x) = 10(0.9999)
(10x) = 9.999
(10x)-x = (9.999) - (0.9999)
9x = 8.9991
x = 0.9999

Ok, so since everybody already thinks that 0.9999.. = 1, there's no point trying to add proof that it is. However, is anyone out here brave enough to try prove that 0.9999... != 1

Sep 7, 2009
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Sep 22, 2010
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4. ### retched AAC Fanatic!

Dec 5, 2009
5,201
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Who is going to be the one to say who is right and who is not?

In a previous thread, you were wrong in your graphic representation and measurement thereof, and you never acknowledged it.

You only went back and edited your previous posts to try to cover your errors.

If you want a serious discussion, it will require a leap of faith on our part.

Is this a topic you pulled out of a hat, or is there a particular problem that led you to post this?

5. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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This came up from an infinity discussion with my friend. He said that infinite number of digits equals to finite number of digits. Which is the 0.999...=1 debate. So he just proved that "infinity" which we can only approximate, is actually a finite number. And right now, I'm trying to prove otherwise.

6. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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Also, regarding the graphic representation of pie - I've never seen the error in the second picture. You've suggested that I should overlap the points - which I think is not practical at all.

7. ### tom66 Senior Member

May 9, 2009
2,613
218
Find me a number between 0.(9) and 1 and you win a prize.

But, you can't.

There is no number between 0.(9) and 1; they are equal.

(BTW: the notation 0.(9) means 0.9999999... and it's an infinite sequence, in case you haven't seen that before.)

Your argument that you're mixing infinite and finite is erroneous, because that happens all the time, anyway. For example, there are an infinite amount of numbers between 0.5 and 1 - you wouldn't say that there are a finite amount.

A lot of people get caught up when they try and imagine 0.(9). They imagine it as terminating at some point. It does not terminate, ever. There is no last digit, and in fact using "0.999... = 1" instead of "0.(9) = 1" catches a lot of people out.

Also, your falsification of the first proof:

is incorrect (you dropped the 5 on step 1), but if you replace 4 with 5, it works:

And you've already given away the second falsification as erroneous by your own choice of example: you will see as you add more digits you get infinitely closer to a finite number and x = 1.

8. ### BestFriend Thread Starter Member

Sep 22, 2010
31
0

Anyway, the first counter-proof highlights the flaw in the proof that we dropped an infinite number of "1" like how I dropped the single "5" and replaced it with ellipses [...]. One is not zero thus it should not be dropped.

Thus does this also make 0.9999...9 and 0.9999...8 the same number? (considering that ellipses [...] represents an infinite and unending process of 9s.

Also, I find this is quite interesting since this is the first time I've seen equality defined as having "no number between". And thank you for sharing that.

I'm sorry but I do not follow. Yes, the number would be approaching one (1) but it would never equal it.

It would just look like this: 8.999...1

9. ### tom66 Senior Member

May 9, 2009
2,613
218
Such a number cannot exist; it would make it terminating, and not infinite. It is like saying you have an infinitely deep pit, and a gremlin at the bottom. (I read this in a children's book once and even at the age of 6 it confused me.)

It is one definition of equality. Others include a difference of zero.

It is converging.

x = 0.9999
9x = 8.9991

x = 0.9999999
9x = 8.999991

x = 0.999...
9x = 9

I am not a mathematician (though hope to become one soon), but this problem has been discussed for centuries, and it doesn't matter how you try to put it, 0.(9) = 1. It is an identity, just like (-1)^2 = 1.

10. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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Yet, for me, it makes sense. It's like saying a tube is infinitely long - and we know the beginning and the end but the middle is infinitely long - growing bigger and bigger.

I've got an idea on how to animate it. I'll get back on the animation.

Thanks for this. I've learned two new things from our discussion.

I think it is like limits - it would never exactly match but would approach it.

x = 0.9999
9x = 8.9991

x = 0.9999999
9x = 8.999991

x = 0.999999999
9x = 8.99999991

Or simply put:

x = 0.999999...
9x = 8.9999...1

9x != 9 but approaches 9

I strongly believe that we all have in ourselves a voice that says we are a mathematicians. A mathematician for me is not someone who has a BS Math diploma from an institution. A mathematician for me is someone who appreciates math and is willing to discuss this fascination with his/her peers. And I believe you are one.

And for me, it doesn't matter how long the problem is discussed. Identities are derived from proofs. And if there is a flaw in that proof, then that identity is also flawed.

Aristotle once argued that infinity exist only as potential infinity. If people in his next generation just took his word for it, there might not be interesting problems in math that tortures students and professors alike.

11. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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Oh yeah, just want to say that adding infinity [...] in the middle and having a definite end is something which I didn't just made up. The article on Wikipedia uses it in the latter more complicated proofs. Although I like how the limit proof actually proves that it just approaches one and never touches it (in a graph).

12. ### Kermit2 AAC Fanatic!

Feb 5, 2010
4,166
1,125
I can prove it ...you silly boy you!

The first example given as FALSE isn't. The decimal equivalent of (1/9) was truncated by someone before the infinite eternity of time needed to represent that number in decimal form could elapse. You give me eternity to prove that an infinity of repeated nines after the decimal EQUALS a 1 and the question is answered.

Simple pimple baby

13. ### DonQ Active Member

May 6, 2009
320
14
First, mathematicians don't say that 0.9999... with infinity digits equals one. They say that the limit of 0.999... as the number of digits approaches infinity, that limit approaches one. At no time did you reach infinity, and at no time did things get equal. It's a semantic way of avoiding exactly this type of pseudo mathematic nonsense.

Second, if you start to change the definition of the very concepts that define a science, then the discussion is no longer about that science at all, it is about your new creation, whatever that is. Kind of like replacing an arc with a straight line segment and claiming that pi is not as understood by mathemeticians.

Infinity is not a number, it is a concept that has been defined by the 'science' of mathematics, to be used according to certain rules. If you don't abide by those rules, then you are not talking about infinity as defined in the framework that you claim to be discussing. Infinity has been defined so that a large class of problems have a result that can be presented concisely, but only according to a strict set of rules. One of the rules is that it is what would make the unrepresentable fraction 0.999...('to' infinity) actually equal to 1, if it were possible, but it is not, thus the math-o-babble about limits and approaching...

Third, once a result is valid to a trillion, trillion, trillion magnitudes of 10 past what you will ever need to calculate the arc extended by the smallest sub-atomic particle at the maximum distance realized by the most powerful telescopes, then for all reasonable intents and purposes, that result is exact. In this sense, 0.999... = 1. And if that isn't close enough, you can always make it a trillion powers of 10 closer, maybe even more.

On the other hand, if we are not constrained by the rules of math, then in very many senses, 0.999... does not equal 1.

Example:

Ascii-Hex 0x30 0x2D 0x39 0x39 0x39 0x2D 0x2D 0x2D

certainly does not equal

Ascii-Hex 0x31

But then, that's not mathematics, is it?

tom66 and Rick Martin like this.
14. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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But wikipedia site said this:

And since the symbol 0.999... represents 0.(9) with infinity digits - they in fact say that "0.9999... with infinite digits equals one."

Yes and no. No, because I never made such claims that Pi is not understood by mathematicians. Yes, because PI is not "PI" ~~ "Pi" being a ratio of the number of 2-dimensional points over the diameter of the circle and bounded by Planck's length. Of course, Pi is "Pi" in the sense that they both share the same definition of circumference divided by diameter - so I unintentionally interchanged the use of Pi and "Pi". Although I just realize that I should have kept the use of the word Pie instead of "Pi" so people would not get confuse. However, the argument is still valid in that "circumference divided by diameter" will sometimes not equal "circumference divided by diameter" - depending on how we define "circumference" and "points" is.

I never said otherwise. Where did I say that Infinity is a number? Also, the "rule" you are referring to was discovered later than the discovery of the concept of infinity.

No, it is still just an approximation. If we apply that thinking to Pi, then you are agreeing that PI is exactly THAT finite number - once the result is valid to a trillion, trillion, trillion magnitudes of 10.

There's a painting, that I greatly admired, which abstractly portrays the concept of exactness. It is entitled: "Ceci n'est pas une pipe"

[/QUOTE]

It depends on the one defining what mathematics is or is not. But I do appreciate your clever way of coming up with a proof. Just that let us not get into that kind of "mathematics" if you'd like to call it one. I think not everyone would agree and understand the validity of that "proof" which is not exactly what the problem pertains to. However, I'd have the pleasure discussing this with you.

15. ### retched AAC Fanatic!

Dec 5, 2009
5,201
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Did you just say:

There is your first problem. Do you know what wikipedia is?

YOU can edit a wiki page. Anyone can. You can start one.

There is a lot more to the structure of what you are avoiding or DONT KNOW because it ISN'T on wikipedia. Grab a book.

16. ### R!f@@ AAC Fanatic!

Apr 2, 2009
9,666
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OP is worse than loosewire

17. ### Georacer Moderator

Nov 25, 2009
5,173
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Dady! Dady! Can I have an Offtopic just for myself? Can I? Can I?

18. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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Yeah? What book exactly?

19. ### BestFriend Thread Starter Member

Sep 22, 2010
31
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What is wrong with you? <snip>

Last edited by a moderator: Oct 17, 2010
20. ### beenthere Retired Moderator

Apr 20, 2004
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293
The OP is once again reminded that an actual "infinity" can and does not exist. As has been pointed out several times, infinity is a concept, useful in math.

Trying to somehow use examples to change infinity from a concept to a reality can't possibly work. For one thing, it is trivial to demonstrate that there are infinitely many different sized infinities. One size does not fit all.

The discussion is starting to become quite unproductive.