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Author Topic:   0.99999~ = 1 ?
InGodITrust
Member (Idle past 946 days)
Posts: 53
From: Reno, Nevada, USA
Joined: 05-02-2009


Message 76 of 237 (543601)
01-19-2010 2:35 PM


Asymtotes
I'm don't have much knowledge of mathematics, but I do remember being taught that an asymtote (which of course is the curve of a function like y=1/x) never reaches a line that it approaches. For example, if y=1/x, then as x approaches infinity, y approaches zero. The curve approaches, but never touches, the x-axis.

What I'm getting from this thread is that 0.9999~ does in fact equal 1. Then similarly, does the asymtote does actually reach the line it approaches?

IGIT

Edited by InGodITrust, : No reason given.

Edited by InGodITrust, : No reason given.

Edited by InGodITrust, : No reason given.

Edited by InGodITrust, : No reason given.


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Dr Jack
Member (Idle past 1382 days)
Posts: 3507
From: Leicester, England
Joined: 07-14-2003


Message 77 of 237 (543602)
01-19-2010 2:51 PM
Reply to: Message 76 by InGodITrust
01-19-2010 2:35 PM


Re: Asymtotes
0.999999~ is the asymptote, not the value approaching it.

To clarify the nomenclature: a curve is said to be asymptotic to a line if it approaches it without ever reaching it (i.e. 1/x is assymptotic to 0), that line itself is the assymptote.

0.9 0.99 0.999 0.9999 0.99999 etc. is a sequence, that sequence approaches but never reaches 1, in a way that resembles an assymptotic curve. 0.999~ is not a member of that sequence, it is its limit. That is, it's the value that 0.9 0.99 0.999 0.9999 0.99999 etc. approaches but never reaches.

Edited by Mr Jack, : Additional


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InGodITrust
Member (Idle past 946 days)
Posts: 53
From: Reno, Nevada, USA
Joined: 05-02-2009


Message 78 of 237 (543603)
01-19-2010 2:57 PM
Reply to: Message 77 by Dr Jack
01-19-2010 2:51 PM


Re: Asymtotes
Okay, I worded that wrong, Mr Jack. What I meant to ask was if, in the example of y=1/x, the curve touches the x-axis.

Thanks
IGIT

Edit---I responded to your last post before you had added to it, so disregard the question above. And thanks for the reply.

IGIT

Edited by InGodITrust, : No reason given.

Edited by InGodITrust, : No reason given.


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Straggler
Member (Idle past 246 days)
Posts: 10332
From: London England
Joined: 09-30-2006


Message 79 of 237 (543604)
01-19-2010 3:02 PM
Reply to: Message 77 by Dr Jack
01-19-2010 2:51 PM


Re: Asymtotes
0.999999~ is the asymptote, not the value approaching it.

I think you have hit the nail on the head here. And IGIT has to be commended for asking the right question.

Even if we think of 0.999R as the infinitie series 9/10 + 9/100 + 9/1000 + 9/10000 + ...... as is the correct way to think of this then I think in our heads we intuitively do the conceptual equivalent of plotting y as the sum of the series and x as the number of terms in the series. In which case y never actually equals 1.

But as you say 0.999R isn't the sum of the series as such. It is the asymptotic value.

It all makes more sense to me expressed like that anyway.


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Rrhain
Member (Idle past 1149 days)
Posts: 6349
From: San Diego, CA, USA
Joined: 05-03-2003


Message 80 of 237 (544065)
01-23-2010 4:45 AM
Reply to: Message 1 by Huntard
01-15-2010 8:19 AM


Yes, as many have pointed out, .999... = 1 exactly. There is no variation between the two of any kind.

This is not a problem for mathematics and, in fact, an extremely significant proof relies upon this notation: That the set of Reals is uncountable.

Suppose the Reals are countable. If so, then you can list them since they would be in one-to-one correspondance with the set of Countable numbers (1, 2, 3, ...). Suppose you had a list of all the Real numbers between 0 and 1, inclusive. Write them out using the repeated-decimal formation where we write .999... for 1. This also means that for any decimal that stops, we use a repeating 9 decimal so that instead of 0.5, we should write 0.4999....

The reason why we do this is that it guarantees that we have a unique representation of each number. No number terminates but instead continues on, even if it is with a bunch of 9s. Thus, we can ensure that two numbers are identical if and only if they are exactly the same at every single decimal place.

So, we'll have a list of decimals:

a1 = 0.a11a12a13...
a2 = 0.a21a22a23...
a3 = 0.a31a32a33...
.
.
.

Now, construct b:

b = 0.b1b2b3...

Where bn = ann such that:

If ann = 1, then bn = 2.
If ann <> 1, then bn = 1.

Thus, it is clear that b <> an for all n. They differ precisely at the nth decimal place.

And yet, b is a number between 0 and 1, inclusive. Therefore, the number of numbers between 0 and 1 is uncountable and thus, the size of the Reals is also uncountable.


Rrhain

Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

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Iblis
Member (Idle past 3172 days)
Posts: 663
Joined: 11-17-2005


Message 81 of 237 (544066)
01-23-2010 5:12 AM
Reply to: Message 1 by Huntard
01-15-2010 8:19 AM


and now for something completely different
For those of you who like this sort of thing, but need something a bit more substantial than "a one by any other name"; give it up for Infigers !!! The rest of that thread will be a bit of a deja vu.

As for the current problem, here's what it looks like for those of us who do our math in our heads.

1/3 = .333...

ergo

1/1 = .999...

qed


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Dr Jack
Member (Idle past 1382 days)
Posts: 3507
From: Leicester, England
Joined: 07-14-2003


Message 82 of 237 (544072)
01-23-2010 6:22 AM
Reply to: Message 80 by Rrhain
01-23-2010 4:45 AM


It's not necessary to write them as 0.9999~ for that proof to work, concluding with an infinite series of 0s also works.

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Rrhain
Member (Idle past 1149 days)
Posts: 6349
From: San Diego, CA, USA
Joined: 05-03-2003


Message 83 of 237 (544088)
01-23-2010 9:52 AM
Reply to: Message 82 by Dr Jack
01-23-2010 6:22 AM


Mr Jack responds to me:

quote:
It's not necessary to write them as 0.9999~ for that proof to work, concluding with an infinite series of 0s also works.

Then how do you distinguish between "0.5000..." and "0.4999..."? The proof rests upon having a decimal expansion and we need to make sure that we don't skip anything or introduce paradox where a number written one way triggers something that it wouldn't have had it been written the other way.

I admit, it's been a while since my Real Analysis class where this was brought up, but that's how I remember the proof: By enforcing a standardized notational format, we can ensure that the process looks at each number individually and we don't overlook something simply because it wasn't notated correctly.


Rrhain

Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Minds are like parachutes. Just because you've lost yours doesn't mean you can use mine.

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Apothecus
Member (Idle past 1687 days)
Posts: 275
From: CA USA
Joined: 01-05-2010


Message 84 of 237 (544104)
01-23-2010 6:03 PM
Reply to: Message 72 by Dr Jack
01-18-2010 1:49 PM


Re: 0 and Infinitesimally Small - Something and Nothing
Hey Mr. Jack.

All real numbers are limits

I've been following this thread for a while now, and there's something stuck and rattling around inside my head. Originally, I had no problem comprehending that .9999... was equal to 1 (you need calculus to get through pharmacy school for some insanely stupid reason) but then I got to thinking about all the other numbers this applies to. Obviously, an infinite number of them, no?

Sorry if this is super obtuse to some of you. I have no wish to waste anyone's time but logically it perplexes me as much as Zeno does. So any number which is represented with an infinite string of trailing nines is equal to the next number, correct? e.g. 3.11239999999..... is equal to 3.11234? 54,433.22223342999999.... is equal to 54,433.22223343? And so on and so on and so on, forever and ever?

Is this right?

If so, then in my mathematically rudimentary skull, infinity just got a lot more infinite.

Have a good .99999999......


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lyx2no
Member (Idle past 3993 days)
Posts: 1277
From: A vast, undifferentiated plane.
Joined: 02-28-2008


Message 85 of 237 (544105)
01-23-2010 6:49 PM
Reply to: Message 84 by Apothecus
01-23-2010 6:03 PM


Re: 0 and Infinitesimally Small - Something and Nothing
3.11239999999..... is equal to 3.11234?
Is a bit off, but…
Is this right?
It is for the rest of it.
Have a good .99999999......

You 1.9999…


You are now a million miles away from where you were in space-time when you started reading this sentence.

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bluescat48
Member (Idle past 3466 days)
Posts: 2347
From: United States
Joined: 10-06-2007


Message 86 of 237 (544107)
01-23-2010 7:04 PM
Reply to: Message 84 by Apothecus
01-23-2010 6:03 PM


Re: 0 and Infinitesimally Small - Something and Nothing
Sorry if this is super obtuse to some of you. I have no wish to waste anyone's time but logically it perplexes me as much as Zeno does. So any number which is represented with an infinite string of trailing nines is equal to the next number, correct? e.g. 3.11239999999..... is equal to 3.11234? 54,433.22223342999999.... is equal to 54,433.22223343? And so on and so on and so on, forever and ever?

Is this right?

Yes as would the quantity 999.99999.... would equal 1000.00000....

the same can be seen in the point that 1/7 + 6/7 = 7/7 = 1. Decimally .(142857).... + (857142).... yields .999999....

This works for any repeating decimal where 1/n + (n-1)/n = n/n

whereas the decimal equivalent of 1/n being a repeating decimal that repeats any group of numbers ie .142857 it's additive inverse when added to the number gives repeating 9's. since 1/7 + its additive inverse, 6/7 = 1 and its decimal equivalent being .(142857).... added to its additive inverse .(857142).... = .999999....
therefore .999999.... = 1

Edited by bluescat48, : = where + should have been


There is no better love between 2 people than mutual respect for each other WT Young, 2002

Who gave anyone the authority to call me an authority on anything. WT Young, 1969

Since Evolution is only ~90% correct it should be thrown out and replaced by Creation which has even a lower % of correctness. W T Young, 2008


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Jon
Inactive Member


Message 87 of 237 (544108)
01-23-2010 7:27 PM
Reply to: Message 4 by Son Goku
01-15-2010 9:00 AM


Re: Totally right!
Son Goku writes:

Here is a proof:
1. One of the axioms of real numbers: Between any two distinct real numbers there is a third real number not equal to either. (The density property)

2. Adding two numbers and dividing by two, results in the number half way between them. By (1), if they are distinct, this should be a new number.

3. 0.999999..... + 1 = 1.999999999.....

4. (1.99999.....)/(2) = 0.99999....., you can check this with long division.

5. Hence, the number halfway between 1 and 0.9999.... is 0.99999....

6. By (2) this is in contradiction with (1), hence

7. 1 and 0.999.... are not distinct and are the same number.

Your logic goes like this:

P · "All real numbers have properties X"
P · ".9999| does not have properties X"
C · ".9999| is = 1 and it is false that .9999| ≠ 1"

Somewhere the logical connection is missing, as your conclusion does not follow from your premises; perhaps you can explain it to me better if I misunderstood. So far, it looks like you've just proven .9999| not to be a real number, which seems more a matter of definition than a matter of real-world fact.

So, I am open to believe you; just provide supporting evidence of ".9999| is = 1 and it is false that .9999| ≠ 1"

I await your modification to your proof

Jon


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

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Dr Adequate
Member (Idle past 328 days)
Posts: 16112
Joined: 07-20-2006


Message 88 of 237 (544109)
01-23-2010 7:51 PM
Reply to: Message 87 by Jon
01-23-2010 7:27 PM


Re: Totally right!
Your logic goes like this:

P · "All real numbers have properties X"
P · ".9999| does not have properties X"
C · ".9999| is = 1 and it is false that .9999| ≠ 1"

No, it goes like this:

P · "Any pair of distinct real numbers has property X."
P · "The pair of real numbers .9999| and 1 does not have property X."
C · ".9999| and 1 are not a pair of distinct real numbers."

Edited by Dr Adequate, : No reason given.


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Jon
Inactive Member


Message 89 of 237 (544116)
01-23-2010 9:41 PM
Reply to: Message 88 by Dr Adequate
01-23-2010 7:51 PM


Re: Totally right!
Dr. A writes:

No, it goes like this:

P · "Any two distinct real numbers have property X."
P · "The two real numbers .9999| and 1 do not have property X."
C · ".9999| and 1 are not distinct real numbers."

LOL. Again, you base your conclusion on the premise—though, I give you extra credit for more cleverly disguising it this time—that .9999| is a real number. Your flow can just as easily be reworded as follows (separating the adjectives out):

P · "All numbers DISTINCT (from 1) have property X1
P · "All REAL numbers have property X2
P · "All numbers that are both REAL and DISTINCT have property X (i.e., X1 + X2)1
P · It is false that .9999| has property X (i.e., that it is both REAL and DISTINCT (from 1))
C · It is false that .9999| is not both REAL and DISTINCT (from 1)

You have proven that REAL & DISTINCT (from 1) do not together describe .9999|, but this only proves that .9999| is either not REAL or either not DISTINCT (from 1) at minimum. Occam's razor tells me to only accept the minimum, which means I need only accept one or the other of your (two) conclusions. The problem is that you've given no reason to accept the REALness of .9999| in place of its DISTINCTness (from 1). So, your reasoning, as it stands, still allows the possibility of .9999| being DISTINCT (from 1)—i.e., it does not prove .9999| to not be DISTINCT (from 1), as all the reasoning given thusfar is still valid if .9999| is DISTINCT (from 1) and not REAL, in other words, it satisfies the conclusion that .9999| not have property X. For your reasoning (and mine) to be accepted, only one of the properties has to be false for .9999|, i.e., it either has to be not REAL or has to be not DISTINCT (from 1). So, you need a separate proof that tells us .9999| is REAL, leaving us to accept only DISTINCTness (from 1) as the property that must be false to satisfy your (and my) original (and accepted-by-both-of-us) reasoning.

Thus, my initial challenge, which was for someone to prove .9999| to be a REAL number which, as the logic shows, could NOT be true if DISTINCTness (from 1) were also true. The full proof(s) will look like this:

Proof that .9999| is REAL:
P · [P1 for .9999| being a REAL number]
P · [P2 for .9999| being a REAL number]
etc.
C · [Conclusion that .9999| is a REAL number]

Proof that .9999| is not DISTINCT (from 1):
P1 · "It is true that 0.9999| is REAL"
P2 · "It is true that 0.9999| is DISTINCT (from 1)"
P3 · "It is false that both P1 and P2 are true (~(P1·P2))"
P4 · "[conclusion from proof of 0.9999| being a REAL number], i.e., P1 is true"
C · "P2 is false, i.e., .9999| is not DISTINCT (from 1)"

Note that premises 1, 2, and 3 are just an expanded restatement from the first proof (beginning of post), with premise 4 (0.9999|'s REALness) being the one we need to insert to make the conclusion that .9999| is not DISTINCT (from 1) unavoidable.

In short, following your (and my) reasoning, there is no way to prove .9999| to be equal to 1 without first showing it to be a real number. Of course, I have no idea if .9999| is a real number or not, and look forward to anyone who will prove that it is and thus prove .9999| = 1. (I just pray to God that the definition that gets used for 'real' number be not so pitiful that anything qualifies.)

Jon

{ABE: I noticed you edited your post after I started typing my reply; worry not, though; italicizing your conclusion doesn't affect its truth value}
__________
1 X1 and X2 are, of course, not necessarily separable properties, but are used to show the necessarily inseparable relationship of the first two premises to the third (i.e., property X only appears when properties X1 and X2 have both been satisfied). All three premises could be rewritten as: "When REAL and DISTINCT (from 1) are both True, then X is false", which would require the addition of "X is True", and the conclusion "REAL and DISTINCT (from 1) are not both True (when said of .9999|)". The problems with this that follow would all still be the same.

Edited by Jon, : LOL


[O]ur tiny half-kilogram rock just compeltely fucked up our starship. - Rahvin

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lyx2no
Member (Idle past 3993 days)
Posts: 1277
From: A vast, undifferentiated plane.
Joined: 02-28-2008


Message 90 of 237 (544119)
01-23-2010 10:24 PM
Reply to: Message 89 by Jon
01-23-2010 9:41 PM


Re: Totally right!
To prove that 0.999… is a real number one reads the dictionary. But fine, if that's the way you want it:
  1. P1[for all real numbers| fits definition of real number.]
  2. P2[0.9999…| fits definition of real number]
  3. C[0.999… is a real number.]


You are now a million miles away from where you were in space-time when you started reading this sentence.

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