
QuickSearch
Welcome! You are not logged in. [ Login ] 
EvC Forum active members: 65 (9073 total) 
 
MidwestPaul  
Total: 893,345 Year: 4,457/6,534 Month: 671/900 Week: 195/182 Day: 28/47 Hour: 0/3 
Announcements:  Security Update Released 
Thread ▼ Details 
Member (Idle past 1535 days) Posts: 2870 From: Limburg, The Netherlands Joined: 

Thread Info



Author  Topic: 0.99999~ = 1 ?  
InGodITrust Member (Idle past 909 days) Posts: 53 From: Reno, Nevada, USA Joined: 
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 xaxis.
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.


Dr Jack Member (Idle past 1345 days) Posts: 3507 From: Leicester, England Joined: 
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


InGodITrust Member (Idle past 909 days) Posts: 53 From: Reno, Nevada, USA Joined: 
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 xaxis.
Thanks EditI 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.


Straggler Member (Idle past 208 days) Posts: 10332 From: London England Joined: 
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.


Rrhain Member (Idle past 1112 days) Posts: 6349 From: San Diego, CA, USA Joined: 
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 onetoone 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 repeateddecimal 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: a_{1} = 0.a_{11}a_{12}a_{13}... Now, construct b: b = 0.b_{1}b_{2}b_{3}... Where b_{n} = a_{nn} such that: If a_{nn} = 1, then b_{n} = 2. 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. 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.


Iblis Member (Idle past 3135 days) Posts: 663 Joined: 
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


Dr Jack Member (Idle past 1345 days) Posts: 3507 From: Leicester, England Joined: 
It's not necessary to write them as 0.9999~ for that proof to work, concluding with an infinite series of 0s also works.


Rrhain Member (Idle past 1112 days) Posts: 6349 From: San Diego, CA, USA Joined: 
Mr Jack responds to me:
quote: 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.


Apothecus Member (Idle past 1650 days) Posts: 275 From: CA USA Joined: 
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......


lyx2no Member (Idle past 3956 days) Posts: 1277 From: A vast, undifferentiated plane. Joined: 
You 1.9999… You are now a million miles away from where you were in spacetime when you started reading this sentence.


bluescat48 Member (Idle past 3429 days) Posts: 2347 From: United States Joined: 
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 + (n1)/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.... 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


Jon Inactive Member 
Your logic goes like this: P · "All real numbers have properties X" 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 realworld 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 halfkilogram rock just compeltely fucked up our starship.  Rahvin


Dr Adequate Member (Idle past 291 days) Posts: 16112 Joined: 
No, it goes like this: P · "Any pair of distinct real numbers has property X." Edited by Dr Adequate, : No reason given.


Jon Inactive Member 
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 X_{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 acceptedbybothofus) 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: Proof that .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} Edited by Jon, : LOL [O]ur tiny halfkilogram rock just compeltely fucked up our starship.  Rahvin


lyx2no Member (Idle past 3956 days) Posts: 1277 From: A vast, undifferentiated plane. Joined: 
To prove that 0.999… is a real number one reads the dictionary. But fine, if that's the way you want it:
You are now a million miles away from where you were in spacetime when you started reading this sentence.



Do Nothing Button
Copyright 20012018 by EvC Forum, All Rights Reserved
™ Version 4.1
Innovative software from Qwixotic © 2022