.9999... equals 1?

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Masterchef

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.9999... equals 1?
« on: March 09, 2007, 05:28:29 PM »
There are a few idiots on a different forum trying to claim that .9999... is actually equal to 1, which anyone with half a brain knows is false. There is no way that any number can equal a different number. Please, can anyone who isn't a fucking moron back me up on this?

http://www.kirupa.com/forum/showthread.php?p=2080661&posted=1#post2080661
« Last Edit: March 09, 2007, 05:35:31 PM by Masterchief2219 »

Re: 1 = .9999... ?
« Reply #1 on: March 09, 2007, 05:36:21 PM »
1 ≈ 0.999∞

It isn't equal, but it is close enough to not matter to anything human. We are finite creatures, hence have a margin of error that can be seen as "correct" and be basically correct; nothing is absolute, fact & truth is relative. So 1=1 is a myth in reality, as there is never exactly "1"; it is always approximate. This means to us, finite beings, an infinitely small difference (as that between 1 and 0.999∞) makes no difference at all, therefore they can be seen as the same.

Basically: correct, they do not equal each other, but it is pointless, and this is a stupid question spawned from boredom at having nothing better to do.  ;)

And the proof that I've seen for it is flawed in one key area:
9.999∞ - 0.999∞ ≠ 9

You can't subtract or add or anything infinite quantities, as that is applying a rule of finite math to infinite. They are basically untouchable in objective mathematics. Saying that "9.9∞ - 0.9∞ = 9" is like saying  "5.5∞*∞ - 4.4∞*∞ = 1.1∞*∞"; it is retarded ... you can't apply finite rules of mathematics to infinite quantities. Its the same thing as dividing a number by zero: if you think it results in realistic mathematics, you are retarded.

∞ - ∞ =
9.999∞ - 0.999∞ =
X∞ - Y∞ =

Simple.
« Last Edit: March 09, 2007, 05:47:57 PM by Erebos »
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Ubuntu

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Re: .9999... equals 1?
« Reply #2 on: March 09, 2007, 05:46:29 PM »
Masterchief2219, I'm sorry, you're wrong this time.

Quote from: Wikipedia
In mathematics, the recurring decimal 0.999 denotes a real number. Notably, this number is equal to 1. In other words, "0.999" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

http://en.wikipedia.org/wiki/0.999

This issue is dealt with extensively here as well: http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html




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Masterchef

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Re: .9999... equals 1?
« Reply #3 on: March 09, 2007, 05:55:43 PM »
How is that even possible?

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Ubuntu

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Re: .9999... equals 1?
« Reply #4 on: March 09, 2007, 05:58:43 PM »
How is that even possible?

They are the same number. 1/2 = 0.5 for the same reason 1 = 0.999...

If they are different numbers, name a number in between them.

1/3 = 0.333...
+2/3 = 0.666...
=3/3 = 0.999...
« Last Edit: March 09, 2007, 06:42:40 PM by Ubuntu »

Re: .9999... equals 1?
« Reply #5 on: March 09, 2007, 05:59:43 PM »
Masterchief2219, I'm sorry, you're wrong this time.

Quote from: Wikipedia
In mathematics, the recurring decimal 0.999 denotes a real number. Notably, this number is equal to 1. In other words, "0.999" represents the same number as the symbol "1". Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

http://en.wikipedia.org/wiki/0.999

This issue is dealt with extensively here as well: http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html


Wow, a completely false wikipedia article. Interesting. No, he is correct. These proofs are bullshit, basically going in circles with obvious gaps in logic (if it can be called logic).



0.333∞ = 1/3 since when? It's almost as if these morons got a first grade calculator and typed in the digits to make up this proof.
0.333∞ ≈ 1/3! Approximately equals! That means there is an infinitely small difference, but there is a difference.

So rewording this "proof" to be correct:
0.333∞ ≈ 1/3
3 x 0.333∞ ≈ 3 x 1/3 ≈ (3 x 1)/3
0.999∞ ≈ 1

Wow, so basically we just gave a bunch of examples of what the approximately equal sign is for ...




This one goes wrong with the:
10c - c = 9.999∞ - 0.999∞
9c = 9

You simply cant subtract 0.999∞ from 9.999∞. Just as ∞ - ∞ ≠ 0, 9.999∞ - 0.999∞ ≠ 9.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #6 on: March 09, 2007, 06:02:42 PM »
How is that even possible?

They are the same number. 1/5 = 0.5 for the same reason 1 = 0.999...

If they are different numbers, name a number in between them.

1/3 = 0.333...
+2/3 = 0.666...
=3/3 = 0.999...

Um, 1/5 = 0.5? I don't think I really even need to comment here.  :p

And a number between them? That is like saying "God exists! If he doesn't, prove it." You can't make a positive claim without support. Anyway, a number between them would be 0.999∞. Learn to comprehend infinite quantities.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #7 on: March 09, 2007, 06:05:46 PM »
Did I win yet?
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Masterchef

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Re: .9999... equals 1?
« Reply #8 on: March 09, 2007, 06:06:46 PM »
The entire argument for this all comes down to the human mind not being able to comprehend infinity.

Re: .9999... equals 1?
« Reply #9 on: March 09, 2007, 06:11:16 PM »
The entire argument for that all comes down to the human mind not being able to comprehend infinity.

Nah, it's not having to do with comprehend. Math simply isn't infinite. We can comprehend the concept of infinity (we simply can't grasp its scope), but we can't deal in it. Math can't deal with infinite quantities, as it is not a finite science; therefore 9.999∞ - 0.999∞ = (no solution). We can comprehend that 1 is different from 9.999∞ by an infinitely small quantity, we simply can't use that quantity for anything as we are not capable of dealing in the infinite. We can represent the quantity with something like 0.0...1, just simply can't use or deal with it in any manner that makes sense. We're dealing with apples & wrenches here, two different things entirely, and not compatible; therefore the proofs are retarded as they pretend we can logically deal in infinite quantities.
« Last Edit: March 09, 2007, 06:12:56 PM by Erebos »
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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Masterchef

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Re: .9999... equals 1?
« Reply #10 on: March 09, 2007, 06:13:25 PM »
Yeah, I actually thought of that after I posted that. :P

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EnragedPenguin

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Re: .9999... equals 1?
« Reply #11 on: March 09, 2007, 06:38:54 PM »

This one goes wrong with the:
10c - c = 9.999? - 0.999?
9c = 9

You simply cant subtract 0.999? from 9.999?. Just as ? - ? ? 0, 9.999? - 0.999? ? 9.

And why not? Surely you must agree that .999... is equal to some number, correct? And that if you subtract that number from .999... you will have 0, correct? And therefore if you subtract that particular number from 9.999... you will have 9. It's the same as subtracting, say, .5 from 2.5. Surely you agree that subtraction would leave 2?
A different world cannot be built by indifferent people.


Re: .9999... equals 1?
« Reply #13 on: March 09, 2007, 06:42:06 PM »

This one goes wrong with the:
10c - c = 9.999? - 0.999?
9c = 9

You simply cant subtract 0.999? from 9.999?. Just as ? - ? ? 0, 9.999? - 0.999? ? 9.

And why not? Surely you must agree that .999... is equal to some number, correct? And that if you subtract that number from .999... you will have 0, correct? And therefore if you subtract that particular number from 9.999... you will have 9. It's the same as subtracting, say, .5 from 2.5. Surely you agree that subtraction would leave 2?

You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999∞ equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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EnragedPenguin

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Re: .9999... equals 1?
« Reply #15 on: March 09, 2007, 06:51:55 PM »
You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999? equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?

.999... does not equal infinity. It may be an infinite string of symobols, but it still represents a finite amount. Any special rules that may apply to adding, subtracting, multiplying, or dividing infinity do not apply to a finite amount.
A different world cannot be built by indifferent people.

Re: .9999... equals 1?
« Reply #16 on: March 09, 2007, 07:00:10 PM »
You're applying laws of finite math to infinite. There is no concept of "equals" in infinity, that is a finite concept used by finite creatures in a finite science. It doesn't matter if .999? equals anything, such a concept doesn't apply to a number with an infinite aspect; the infinite aspect makes it untouchable by finite mathematics.

Do you think that infinity = infinity?

.999... does not equal infinity. It may be an infinite string of symobols, but it still represents a finite amount. Any special rules that may apply to adding, subtracting, multiplying, or dividing infinity do not apply to a finite amount.

It has an infinite aspect; there are no "special rules" per say, it's basically just mathematic common sense. It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

9.999... - .999... simply does not equal 9

It has no solution.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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EnragedPenguin

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Re: .9999... equals 1?
« Reply #17 on: March 09, 2007, 07:07:31 PM »
It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

But .999... represents neither an infinitely small quantity nor an infinitely large quantity. It represents a definite, finite quantity, and can therefore be treated as such.
Just as 2+2, 8-4, 164, (?4)^2 all represent 4, .999... is just another way of representing 1.
A different world cannot be built by indifferent people.

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skeptical scientist

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Re: .9999... equals 1?
« Reply #18 on: March 09, 2007, 07:12:25 PM »
Masterchief, are you serious, or are you joking? The identity .99999...=1 is a well known fact. Every so often people doubt it, but such people don't even know the definitions.

First question: what is meant by the representation .98604, or .33333..., or .9999....?

Answer: each of these is shorthand for a sum, where the nth digit is the multiple of 10-n that we are adding. So .98604 is simply shorthand for 9*10-1+8*10-2+6*10-3+0*10-4+4*10-5, and .3333... is shorthand for 3*10-1+3*10-2+3*10-3+3*10-4+....

This immediately raises a second question: what is meant by an infinite sum? You can't add infinitely many things, can you?

Answer: yes, you can, in some situations. You can add 1+0+0+0+0+0+..., because after the first term you aren't changing things at all. You can add 1/2+1/4+1/8+1/16+..., because such a sum can be represented "physically" on the number line as follows: first you go half way from 0 to 1, then you go half way from where you are to 1, then you go half way from where you are now to 1, etc. The sum is 1, because that's exactly where you get when you move half way to one infinitely many times (this is not, of course, a proof, but acts as motivation for the idea of limits.) You cannot, however, take the sum 1-1+1-1+1-..., because that sum alternates between 1 and 0, and never settles on any one number (anyone who tells you that sum exists is either completely nuts or has been staring at the zeta function too long). The definition that makes this work is the definition of limits. A sequence of numbers x_n is just that: a sequence x_1, x_2, x_3, x_4, .... Such a sequence is said to converge to a limit x if for any e>0, there is an N>0 so that the differences between x and x_n are less than e for n>N. An infinite sum is just the limit of the partial sums of the initial terms, provided that limit converges. So the decimal number .999999... really is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc.

They don't teach you all this when they teach you decimals in school, because it's complicated, so they usually just gloss over it, but it is what decimal representations actually are, and so any demonstration that .999...=1 has to use this fact (of course, there are some tricks people sometimes use to "cheat", but those involve taking things on faith, which are true, but which are exactly the things that Erebos is denying above). The lack of such actual definitions is why so many people are confused, or try to deny that .9999...=1. But what's true is that decimal numbers are really just representations of real numbers, and are not always unique. The fact that 1 can also be written as .9999.... is just one example.

So, without further ado-

Proof that 1=.99999...:
We know that .999... is, by definition, the sum 9/10+9/100+9/1000+... which is the limit of the sums 9/10, 9/10+9/100, 9/10+9/100+9/1000, etc. Simply finding common denominators tells us that 9/10+9/100+9/1000+...+9/10n=99...9/10n=(10n-1)/10n=1-1/10n, and since for any e>0, 1/10n is eventually smaller than e, this limit is 1.

p.s. I decided to change my avatar in honor of this thread.
« Last Edit: March 09, 2007, 07:24:27 PM by skeptical scientist »
-David
E pur si muove!

Re: .9999... equals 1?
« Reply #19 on: March 09, 2007, 09:21:40 PM »
Sigh. I'd like to know why the fuck there is an "approximately equal" sign in math, then; because apparently it has no use, now that something not equal to one (but approximately equal) miraculously now "equals" one.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


Re: .9999... equals 1?
« Reply #20 on: March 09, 2007, 09:36:17 PM »
It is an infinite number in the same way that an infinitely small number is an infinite number, and it is the same as if it was an infinite quantity in regards to applying finite mathematics.

But .999... represents neither an infinitely small quantity nor an infinitely large quantity. It represents a definite, finite quantity, and can therefore be treated as such.
Just as 2+2, 8-4, 164, (?4)^2 all represent 4, .999... is just another way of representing 1.
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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skeptical scientist

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Re: .9999... equals 1?
« Reply #21 on: March 09, 2007, 09:41:58 PM »
Sigh. I'd like to know why the fuck there is an "approximately equal" sign in math, then; because apparently it has no use, now that something not equal to one (but approximately equal) miraculously now "equals" one.
There isn't an "approximately equal" sign in math. It only exists in physics, statistics, and other subjects where precision in definitions is frequently given up for the sake of stating things simply. In math, people are careful to define what they mean by an approximation. For example, a physicist would say,

or even just nn/en, and leave it at that. A mathematician, on the other hand would say that,
, and might additionally mention the bounds on the ratio, and the rate at which the ratio converges.

You are completely wrong in saying, "something not equal to one (but approximately equal) miraculously now 'equals' one". As I have explained, the real number represented by .999... is exactly the same as the number represented by 1. There is no approximate equality here; the two quantities are identical.
« Last Edit: March 09, 2007, 09:50:14 PM by skeptical scientist »
-David
E pur si muove!

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skeptical scientist

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Re: .9999... equals 1?
« Reply #22 on: March 09, 2007, 09:42:49 PM »
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
-David
E pur si muove!

Re: .9999... equals 1?
« Reply #23 on: March 09, 2007, 09:48:41 PM »
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.

Go figure; did I ever say it was a "real" number?
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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skeptical scientist

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Re: .9999... equals 1?
« Reply #24 on: March 09, 2007, 09:49:41 PM »
If decimals don't represent real numbers, what the hell are they?
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Dioptimus Drime

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Re: .9999... equals 1?
« Reply #25 on: March 09, 2007, 09:52:43 PM »
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
I really don't know enough to demand authority on this, but out of curiosity, couldn't you approximate a number as such: 0.000...1
Wouldn't that be an infinitely small real number?

~D-Draw

Re: .9999... equals 1?
« Reply #26 on: March 09, 2007, 09:55:21 PM »
It represents a definite finite quantity minus an infinitely small quantity. It includes the infinitely small quantity.
There is no such thing as an infinitely small real number.
I really don't know enough to demand authority on this, but out of curiosity, couldn't you approximate a number as such: 0.000...1
Wouldn't that be an infinitely small real number?

~D-Draw

According to his reasoning that would equal zero. In math, that is accepted because it makes no difference, in reality it's retarded to say it is actually true.
How? when? and whence? The gods give no reply. Let so it is suffice, and cease to question why.


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beast

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Re: .9999... equals 1?
« Reply #27 on: March 09, 2007, 10:10:22 PM »
I think this topic really just demonstrates to what level people have studied maths.  There is no doubt that in the formal system of mathematics, 0.9 recurring is the same number as 1.  It is not almost the same, it is the same.

I also wanted to comment on this statement;

Quote
Math simply isn't infinite. We can comprehend the concept of infinity (we simply can't grasp its scope), but we can't deal in it. Math can't deal with infinite quantities, as it is not a finite science;

This is completely false.  There are whole fields of maths that deal with infinity and infinity is a crucial concept in maths.  In fact many proofs rely on dealing with infinity.  Your arguments are clearly based out of an ignorance of maths and philosophical understanding, rather than a factual understanding.  I suggest you actually take an advanced maths, and you'll clearly see how wrong you are.

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skeptical scientist

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Re: .9999... equals 1?
« Reply #28 on: March 09, 2007, 10:13:49 PM »
D-Draw, what you have just given me is not a decimal number. Decimal numbers are infinite sequences of digits. What you gave is something that has infinitely many digits, and then after all of those digits, there's another one. One can make sense of such objects, but not as decimal representations of real numbers. You can also define systems where .999... and 1 are not the same object. However, if you look at the actual definition of real number, and the actual definition of a decimal representation, then .999...=1. To see all the definitions in detail, it would be best to read a book on real analysis - such as Rudin's Principles of Mathematical Analysis (Amazon) - or take a class on the subject.

Quote
In math, that is accepted because it makes no difference, in reality it's retarded to say it is actually true.
It's not accepted because it "makes no difference", it's accepted because it follows from the definitions of real number and decimal representation. That makes it true. If you refuse to accept it, then you either mean something completely different by decimal than the rest of the world means (including all of the educated mathematicians), or else you're simply clueless.
-David
E pur si muove!

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beast

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Re: .9999... equals 1?
« Reply #29 on: March 09, 2007, 10:49:25 PM »
Here are 4 maths based websites that all claim that 0.9r = 1.  I challenge all those who do not believe that this is the case to present a genuine maths based website with evidence supporting your beliefs.


http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/
http://mathforum.org/library/drmath/view/57035.html
http://sprott.physics.wisc.edu/Pickover/pc/9999.html
http://www.purplemath.com/modules/howcan1.htm