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.999.... (Repeating) is equal to one?

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RavenWilder Since: Apr, 2009
#126: Sep 10th 2011 at 6:57:04 AM

I'm gonna repeat myself:

Since the nines go on forever, the difference between .999... and 1 is infinitely small. And if "infinitely large" encompasses everying, than "infinitely small" encompasses nothing. So "nothing" is the difference between those two numbers.

Yej (4 Score & 7 Years Ago) Relationship Status: They can't hide forever. We've got satellites.
#127: Sep 10th 2011 at 6:57:18 AM

[up][up] To confuse things further, for an arbitarary real number, it's impossible to construct the largest number that is smaller than it. (or the smallest number that is larger.)

[up] No, "infinitely small" is not zero, just as close as can be. wink

edited 10th Sep '11 6:58:09 AM by Yej

DarkConfidant Since: Aug, 2011
#128: Sep 10th 2011 at 7:08:29 AM

[up] True, but you can construct a series / sequence that approaches an arbitrary real number, then find the least upper bound of the infinite set constructed by the terms (or by the set of partial sums).

abstractematics Since: May, 2011
#129: Sep 10th 2011 at 7:53:54 AM

In real numbers, you can't have infinitely small. It's a very simple proof by contradiction that there is no smallest real number.

Assume that x is the smallest (positive) real number. However, properties of real numbers dictate that if x is real, then x/2 is also real (and positive). Obviously x/2 < x if x is positive, a contradiction. (We can change x/2 to 2x to prove that there is no largest real number, neither most positive nor most negative).

This is what I meant by infinite divisibility and accuracy.

@Exclamation Mark That's contradictory. If it's exactly equal in theory, then it's definitely "equal" in real world. If we treat 1.01 as "basically" 1 and 3.14 as basically pi, then why shouldn't 1 = 1?

And calculators are designed to be finite. You can't always rely on them to do proofs.

@Swish

Math is not bound by real world. A natural example showing the concept of infinity is the set of integers. No matter how far you count, you're going to have more integers, making the set infinite.

Now using Trivialis handle.
Yej (4 Score & 7 Years Ago) Relationship Status: They can't hide forever. We've got satellites.
#130: Sep 10th 2011 at 8:12:18 AM

Slightly bizarrely, the integers sans any subset of the integers is the same size as the full set of integers.

abstractematics Since: May, 2011
#131: Sep 10th 2011 at 8:12:54 AM

Well that goes back to "infinity - finite amount = infinity" point.

Now using Trivialis handle.
Yej (4 Score & 7 Years Ago) Relationship Status: They can't hide forever. We've got satellites.
#132: Sep 10th 2011 at 8:14:58 AM

No, but it even works when you use infinite subsets. {x E Z | x > 0} == Z.

abstractematics Since: May, 2011
#133: Sep 10th 2011 at 8:21:51 AM

Well yeah, you can't have an infinite set with smaller cardinality than N.

It makes more sense when you throw out the "numbers" idea and treat the elements as just objects.

[down]Which expression?

edited 10th Sep '11 9:02:15 AM by abstractematics

Now using Trivialis handle.
Tzetze DUMB from a converted church in Venice, Italy Since: Jan, 2001
DUMB
#134: Sep 10th 2011 at 8:49:41 AM

I have a tangentially related question. Why is the matter of 0.999... = 1 so darn controversial?

I don't know. This is almost as bad as that ill-formed arithmetic expression that went memetic a few weeks ago.

By the way. I think this thread has lead me to a greater understanding of why Cantor went in and out of mental hospitals.

[1] This facsimile operated in part by synAC.
Enthryn (they/them) Since: Nov, 2010
(they/them)
#135: Sep 10th 2011 at 9:37:48 AM

"Infinitely small" is meaningless in the real numbers. (And expressions like 0.000...01 are nonsensical. You can't have an infinite sequence of digits with 1 at the "end". It'd be like a 4-sided triangle: a contradiction in terms.) If you want to work with weird concepts like that, you have to work with a system that has infinitesimals, such as the hyperreals or another nonstandard number system. At that point, you're not really working with "numbers" in the usual sense.

On the topic of infinity, as I've said before, mathematicians work with things that are actually infinite all the time. Sure, that means you can't represent it physically in certain ways, but you don't need to. If it bothers you that we work with the infinite, too bad; it's quite useful to do so. (The real fun starts when we use infinite sets that are larger than the usual infinity.)

[up][up] Technically, if you don't use the axiom of choice, you can (probably) consistently assert that certain uncountable sets have subcountable cardinality. But if you accept the axiom of choice, you can prove that such subcountable sets don't exist.

edited 10th Sep '11 9:41:40 AM by Enthryn

ekuseruekuseru 名無しさん from Australia Since: Oct, 2009
名無しさん
#136: Sep 10th 2011 at 10:09:18 AM

@Yej

"No, "infinitely small" is not zero, just as close as can be."

The problem here is that "as close as can be" to zero is infinitely close, which, accepting Raven's reasoning, is zero. The statement that "the number that is infinitely small is as close as can be to zero" is true irrespective of what you believe, and is therefore not contradictory to Raven's statement. What you need to do is independently establish why the statement "the number as close as can be to zero is zero" is false.

And I don't think you can.

onyhow Too much adorableness from Land of the headpats Since: Jan, 2001 Relationship Status: Squeeeeeeeeeeeee!
Too much adorableness
#137: Sep 10th 2011 at 10:41:24 AM

I have a tangentially related question. Why is the matter of 0.999... = 1 so darn controversial?

Because it pushes human ability to percieve the world with instincts and "gut feelings" over the limit? Things just get so strange that even with proofs people still just refuse to believe it...or just incapable of imagine it...

edited 10th Sep '11 10:42:39 AM by onyhow

Give me cute or give me...something?
Enkufka Wandering Student ಠ_ಠ from Bay of White fish Since: Dec, 2009
Wandering Student ಠ_ಠ
#138: Sep 10th 2011 at 10:42:52 AM

hehehe... you said "limit"

Very big Daydream Believer. "That's not knowledge, that's a crapshoot!" -Al Murray "Welcome to QI" -Stephen Fry
Yej (4 Score & 7 Years Ago) Relationship Status: They can't hide forever. We've got satellites.
#139: Sep 10th 2011 at 3:41:15 PM

[up][up][up] I mean the smallest, positive non-zero value. This isn't a real number, but it's possible to construct.

Tzetze DUMB from a converted church in Venice, Italy Since: Jan, 2001
Yej (4 Score & 7 Years Ago) Relationship Status: They can't hide forever. We've got satellites.
Ponicalica from facing Buttercup Since: May, 2010
#142: Sep 10th 2011 at 4:24:44 PM

I don't think it's constructible in the surreals.

the future we had hoped for
Tzetze DUMB from a converted church in Venice, Italy Since: Jan, 2001
DUMB
#143: Sep 10th 2011 at 4:36:12 PM

You can construct ε as ω-1, but I think you can also construct arbitrary powers of ε.

[1] This facsimile operated in part by synAC.
INUH Since: Jul, 2009
#144: Sep 10th 2011 at 4:47:27 PM

Two numbers are considered distinct if and only if a number between them exists.

No number between .999 repeating and one exists. Therefore, they are not distinct numbers.

Infinite Tree: an experimental story
TheDeadMansLife Lover of masks. Since: Nov, 2009
Lover of masks.
#145: Sep 10th 2011 at 4:53:59 PM

ill-formed arithmetic expression that went memetic a few weeks ago

Huh?

Please.
Enkufka Wandering Student ಠ_ಠ from Bay of White fish Since: Dec, 2009
Wandering Student ಠ_ಠ
#146: Sep 10th 2011 at 4:55:35 PM

It was something like 1+1+1+1+1+1+1*0=?

remember the order of operators, people!

Very big Daydream Believer. "That's not knowledge, that's a crapshoot!" -Al Murray "Welcome to QI" -Stephen Fry
TheDeadMansLife Lover of masks. Since: Nov, 2009
Lover of masks.
#147: Sep 10th 2011 at 4:58:17 PM

I have never seen that one. Thought you where going to say the 2=1 thing, or whatever it is involving variables.

>.>

The answer is 6 right?

Please.
Enkufka Wandering Student ಠ_ಠ from Bay of White fish Since: Dec, 2009
Wandering Student ಠ_ಠ
#148: Sep 10th 2011 at 4:59:11 PM

its ludicrously easy things like that. No clue how many 1s I put there, but I have seen that on facebook, and a significant amount of people put 0.

>Headdesk<

And the 2=1 thing makes errors because it divides by 0.

edited 10th Sep '11 4:59:53 PM by Enkufka

Very big Daydream Believer. "That's not knowledge, that's a crapshoot!" -Al Murray "Welcome to QI" -Stephen Fry
TheDeadMansLife Lover of masks. Since: Nov, 2009
Lover of masks.
#149: Sep 10th 2011 at 5:02:26 PM

I know why it fails.

>.> It does equal six though right? I just took a math test and I am stuck in the mode where I can't add 1+1 for like five hours with out triple-checking my work.

Please.
Enthryn (they/them) Since: Nov, 2010
(they/them)
#150: Sep 10th 2011 at 5:05:20 PM

There are a few others "proofs" that 2 = 1. One trick is pretending that the complex exponential is injective (which it isn't), and concluding that a^b = a^c (where a > 0 and a ≠ 1) implies b = c. This works for real-valued b and c because the real exponential function actually is injective, but it doesn't work for complex numbers.

edited 10th Sep '11 5:06:22 PM by Enthryn


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