0.99999~ = 1 ?

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.

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

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
IGITEdit---I responded to your last post before you had added to it, so disregard the question above. And thanks for the reply.IGITEdited by InGodITrust, : No reason given.Edited by InGodITrust, : No reason given.

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.

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 b_n = a_nn such that:If a_nn = 1, then b_n = 2.
If a_nn <> 1, then b_n = 1.Thus, it is clear that b <> a_n 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.

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Rrhain

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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.

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

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...ergo1/1 = .999...qed

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

Mr Jack responds to me:

quote:

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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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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.

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Rrhain

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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.

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

Hey Mr. Jack. 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......

Is a bit off, but It is for the rest of it. You 1.9999

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

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/nwhereas 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.... = 1Edited by bluescat48, : = where + should have been

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There is no better love between 2 people than mutual respect for each other WT Young, 2002Who gave anyone the authority to call me an authority on anything. WT Young, 1969Since 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

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

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

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.

LOL. Again, you base your conclusion on the premisethough, I give you extra credit for more cleverly disguising it this timethat .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 (~(P1P2))"
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}
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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

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

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.]

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