## A very interesting Mathematical Paradox

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jestingrabbit
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I've got the lock button right there, and my finger's twitchin'...

That said, the discussion about different long division algorithms has allowed me to fulfill my "learn something new everyday" quotient.

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jestingrabbit wrote:I've got the lock button right there, and my finger's twitchin'...

If you lock it, it will only cause people to create new threads on the same tired topic. It's something of a lose-lose situation for those annoyed at the deniers.
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jestingrabbit
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But then I could just delete those posts as they happen

But seriously, it'll stay open, but everytime I come here I do want to poke my eyes out with a spoon.

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http://tinyurl.com/3copex

Which also brings up an interesting corollary.

0.666666... + 0.5555.... = 1.22222... not 1.111111..

That's kick ass.

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http://tinyurl.com/3copex

Which also brings up an interesting corollary.

0.666666... + 0.5555.... = 1.22222... not 1.111111..

That's kick ass.

...and obvious from the usual addition algorithm? You have to carry the 1.
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ZeroSum
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1/243 is a fun repeating decimal which nicely illustrates the carry the one idea and how it changes the result in a fun way.

mrkite
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skeptical scientist wrote:...and obvious from the usual addition algorithm? You have to carry the 1.

Actually, what I thought was kick ass was that you carried a 1 from a non-existent end point.

The way the usual addition algorithm works is from right to left.. but you can't start at infinity and work your way right..

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mrkite wrote:
skeptical scientist wrote:...and obvious from the usual addition algorithm? You have to carry the 1.

Actually, what I thought was kick ass was that you carried a 1 from a non-existent end point.

The way the usual addition algorithm works is from right to left.. but you can't start at infinity and work your way right..

I don't see how that's unusual...

0.6666666666... = 6/9
0.5555555555... = 5/9

0.66666666... + 0.5555555555... = 6/9 + 5/9

6/9 + 5/9 = 11/9

11/9 = 1.222222222...

Why would you expect to get 1.11111111...?
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mathmagic wrote:
mrkite wrote:
skeptical scientist wrote:...and obvious from the usual addition algorithm? You have to carry the 1.

Actually, what I thought was kick ass was that you carried a 1 from a non-existent end point.

The way the usual addition algorithm works is from right to left.. but you can't start at infinity and work your way right..

I don't see how that's unusual...

0.6666666666... = 6/9
0.5555555555... = 5/9

0.66666666... + 0.5555555555... = 6/9 + 5/9

6/9 + 5/9 = 11/9

11/9 = 1.222222222...

Why would you expect to get 1.11111111...?

Because of this.

8/9 = 0.88888...
9/9 = 0.99999...
10/9 = 1.00000...
11/9 = 1.11111...

*runs*

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.999... = 1.0

QED
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mrkite wrote:
skeptical scientist wrote:...and obvious from the usual addition algorithm? You have to carry the 1.

Actually, what I thought was kick ass was that you carried a 1 from a non-existent end point.

The way the usual addition algorithm works is from right to left.. but you can't start at infinity and work your way right..

No, you start at the first digit you want to calculate, and start from one place to the right of it. You might have to look a bit further if that place may or may not carry depending on whether the place after it carries.

Alternatively, you see the regularity and observe that you can carry all the ones at once, mentally, without having to go through the algorithmic procedure.
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You can add left-to-right, too, you know. Then you get 0.XXXXXX..., where X=11. Usually we write base ten digits, so for the leftmost one, carry the 1 to make it 1.1XXXXXX... Work your way to the right, carrying the 1 over to the left each time and voila! 1.2222222222... Loosely speaking, the only digit after the first that isn't a 2 is "the digit at infinity" (and vis-a-vis epsilon/delta can be ignored).

Seriously, though, doing arithmetic from left-to-right instead of right-to-left has several advantages. Foremost among them is the ability to get a good approximation right off the bat. It's usually more practical to know something is 10 million something than to know the last two digits are 43. Left-to-right does that for you. [/rant]

[Edit: Well, dang. I want to answer the two posts below, but this thread doesn't deserve to be bumped... I know! I'll edit!]
I know there is no "digit at infinity" and I know that's the reason why many people don't get that 0.99999... = 1. My point is that if there was, then for every epsilon > 0, the difference between 0.9999... and 1 is smaller than epsilon. Between any two distinct real numbers there are infinitely many real numbers, but you can't fit anything between 0.99999... and 1, so they aren't distinct.

Basically the same thing with the 1.2222..., but easier to say with 0.99999... (though I think I still didn't say what I meant very clearly).
[/Edit]
Last edited by Nimz on Tue Sep 18, 2007 4:26 pm UTC, edited 1 time in total.
LOWA

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Nimz wrote:You can add left-to-right, too, you know. Then you get 0.XXXXXX..., where X=11. Usually we write base ten digits, so for the leftmost one, carry the 1 to make it 1.1XXXXXX... Work your way to the right, carrying the 1 over to the left each time and voila! 1.2222222222...

That's what I meant when I said, "Alternatively, you see the regularity and observe that you can carry all the ones at once, mentally, without having to go through the algorithmic procedure."

Loosely speaking, the only digit after the first that isn't a 2 is "the digit at infinity" (and vis-a-vis epsilon/delta can be ignored).

Ack! There is no digit at infinity, and it's not ignored because of epsilon/deltas. It just doesn't exist - the definition of decimal notation doesn't include one. This is half the reason so many people think that .999... is different from 1.
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jestingrabbit
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skeptical scientist wrote:
Loosely speaking, the only digit after the first that isn't a 2 is "the digit at infinity" (and vis-a-vis epsilon/delta can be ignored).

Ack! There is no digit at infinity, and it's not ignored because of epsilon/deltas. It just doesn't exist - the definition of decimal notation doesn't include one. This is half the reason so many people think that .999... is different from 1.

Agreed. Do not even speak about "the digit at infinity" loosely.

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mrkite wrote:.9999.... doesn't exist. 0.333333.... is the result of long division. I don't know of any division that could ever result in 0.999999....

I think this is why it's hard to grasp the concept here. In a duodecimal system, 1/3 = 0.4, 2/3 = 0.8, 3/3 = 1 = 0.4+0.8 [8+4=12, which results in 1]. There is no issue there, therefore there should not be an issue in any other radix. It almost seems like, in my mind, that 0.333~*3 would equal 1, right off the bat.

So I think if this 0.999~ number did exist, this infinitesimal, repeating number, that it would not equal 1. I had to respond because no one had mentioned the similarity to the speed of light issue. Scientists always say that there is absolutely no way anyone, in any device, could travel at the speed of light, but they could approach it. The speed of light c = 1, and u = 0.999~. If 0.999~ equaled 1, then we could travel at the speed of light, a scientific impossibility.

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### Re:

azule wrote:
mrkite wrote:.9999.... doesn't exist. 0.333333.... is the result of long division. I don't know of any division that could ever result in 0.999999....

I think this is why it's hard to grasp the concept here. In a duodecimal system, 1/3 = 0.4, 2/3 = 0.8, 3/3 = 1 = 0.4+0.8 [8+4=12, which results in 1]. There is no issue there, therefore there should not be an issue in any other radix. It almost seems like, in my mind, that 0.333~*3 would equal 1, right off the bat.

So I think if this 0.999~ number did exist, this infinitesimal, repeating number, that it would not equal 1. I had to respond because no one had mentioned the similarity to the speed of light issue. Scientists always say that there is absolutely no way anyone, in any device, could travel at the speed of light, but they could approach it. The speed of light c = 1, and u = 0.999~. If 0.999~ equaled 1, then we could travel at the speed of light, a scientific impossibility.

First, why on earth did you necro this thread? I know necro-ing is allowed on these forums, but if you pop up to the Mathematics board you'll note a sticky thread that should tell you what the local opinion on the 0.999... debate is.
Second, 0.999... is not an infinitesimal. An infinitesimal is a type of number that is "really small" similar to how an infinity is "really big". In the real numbers, there is only one infinitesimal, which is zero.
Third, 0.999... is a number, in the sense that it is a valid decimal representation. As such, if you accept that every valid decimal represents a real number, then you can prove that the only real number it can represent is the same one represented by 1, and 1.0, and 1.000... and so forth.
Fourth, you've got your relativity hopelessly wrong but I'm not going to go into that.
Fifth, you're not the first person to compare the two.
Sixth, saying that "this is impossible in science therefore this apparently analogous situation is impossible in mathematics" is not actually a valid argument - many things can be represented in mathematics that have no "real-world" scientific counterpart.
Seventh, sure 1/3 = 0.4 in duodecimal, but in duodecimal you also have the number 0.BBB..., which is 11/11, which is 1. And in fact in base-n for every integer n>1, there is an equivalent n-imal representation 0.(n-1)(n-1)(n-1)... that equals 1, so it actually *is* an issue in other radixes.
Eighth, I strongly recommend you read the Wikipedia article on the subject, which covers it from pretty much every angle you could ever hope for (and it still equals 1).
Ninth, aaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrrggggggggggggggghhhhhhhhhhhhh! I can't believe I got sucked into this discussion again!
Last edited by ConMan on Wed Jul 28, 2010 1:17 pm UTC, edited 1 time in total.
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### Re: Re:

ConMan wrote:First, why on earth did you necro this thread?

I knew I'd get a response like this. See, below.

ConMan wrote:Fourth, you've got your relativity hopelessly wrong but I'm not going to go into that.

I'm guessing this has to do with why it's impossible to reach the speed of light. But yeah, I wasn't making a scientific argument, but a mathematical to real-life connection argument. It was more of a glance at the issue. And no, I'm probably not gonna win any arguments over in a Science Forum either. Lol.

ConMan wrote:Fifth, you're not the first person to compare the two.

Alright. Sorry to whoever laid that comparison out. I must have missed it.

ConMan wrote:Sixth, saying that "this is impossible in science therefore this apparently analogous situation is impossible in mathematics" is not actually a valid argument - many things can be represented in mathematics that have no "real-world" scientific counterpart.

This is why I felt compelled to respond. Math should represent real life. Even if it's just theoretical real life. Maybe I don't know "anything" about math, or science, but they taught me the same as they taught everyone else in school, so there must have been some reason for me to personally know this stuff (mathematical representations) and not just the people going for a Master's.

ConMan wrote:Ninth, aaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrrggggggggggggggghhhhhhhhhhhhh! I can't believe I got sucked into this discussion again!

Sorry, dude.

I figured, someone like you, who'd been there and done that with this thread might just avoid it based on the knowledge that you probably wouldn't find anything new or debatable to you. Others like me, might enjoy having a little convo about it, though.

I'd never heard of this problem before, but I did restrain myself from rehashing any arguments already put forth. And yes, I did read the Wiki link, and a bunch of the other links.

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### Re: Re:

azule wrote:
ConMan wrote:Fourth, you've got your relativity hopelessly wrong but I'm not going to go into that.

I'm guessing this has to do with why it's impossible to reach the speed of light. But yeah, I wasn't making a scientific argument, but a mathematical to real-life connection argument. It was more of a glance at the issue. And no, I'm probably not gonna win any arguments over in a Science Forum either. Lol.

Analogy is fine, but here the analogy works on another level because 0.999... and the speed of light are both things people frequently get wrong - and they both involve limits which are something else people get confused with (here's something useful to remember - a statement S(n) that is true for all integer n is not guaranteed to be true in the limit as n goes to infinity). It is definitely true that under the conditions of special relativity (i.e. in an inertial frame of reference), then an object with non-zero rest mass can never be accelerated up to or past c without somehow applying an infinite amount of energy. Similarly, the terms of the sequence 0.9. 0.99, 0.999, are all strictly less than 1, but the limit of said sequence as the number of decimal places goes to infinity is not bound by the same rules, so that 0.999... is perfectly capable of equalling 1. If you want to follow more about how relativity works, I daresay there's a relevant thread in the Science forum.

azule wrote:
ConMan wrote:Fifth, you're not the first person to compare the two.

Alright. Sorry to whoever laid that comparison out. I must have missed it.

Not necessarily in this thread. I did most of my arguing on this topic over on Wikipedia - specifically on the Talk page and on the associated Arguments page (fun fact - 0.999... was, IIRC, the second Wikipedia page to get an "Arguments" page due to debate on the Talk page drowning out actual discussion over how to improve the article, the first one being "Mohammed"). I'm pretty sure it was brought up there, lost amongst the painfully formatted and nigh incomprehensible ranting.

azule wrote:
ConMan wrote:Sixth, saying that "this is impossible in science therefore this apparently analogous situation is impossible in mathematics" is not actually a valid argument - many things can be represented in mathematics that have no "real-world" scientific counterpart.

This is why I felt compelled to respond. Math should represent real life. Even if it's just theoretical real life. Maybe I don't know "anything" about math, or science, but they taught me the same as they taught everyone else in school, so there must have been some reason for me to personally know this stuff (mathematical representations) and not just the people going for a Master's.

There are a few schools of thought about how mathematics and the real world relate, but the one I subscribe to is based on the idea that mathematics in itself is quite a pure thing, and you work from a basic framework of axioms to build up elegant theorems in various fields. Then science makes some measurements of the real world and looks into the fields of mathematics to find something that can help them put together a model to explain those measurements, which forms part of a theory.

Sometimes the mathematics needs to be developed to encompass the theory, sometimes it's been in place for some time, often just considered a "curiosity" until they realise it's actually useful for something - case in point, non-Euclidean geometry was kind of interesting to see how things might work in a system where, for example, parallel lines meet on a regular basis, and to improve our understanding of what makes Euclidean geometry so special, but otherwise it was just a bit of a nifty trick. Then suddenly General Relativity comes along and this kind of thing can actually be used to describe how our universe is shaped.

I don't blame you for making the connection, given that the maths and science they teach at schools is either completely disconnected, or else somehow explained as being two sides of the same coin, but they're really just two fields (or more, depending on how you cut it) that happen to work really well in tandem, even if they sometimes go off and do their own thing.

azule wrote:
ConMan wrote:Ninth, aaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrrggggggggggggggghhhhhhhhhhhhh! I can't believe I got sucked into this discussion again!

Sorry, dude.

I figured, someone like you, who'd been there and done that with this thread might just avoid it based on the knowledge that you probably wouldn't find anything new or debatable to you. Others like me, might enjoy having a little convo about it, though.

I'd never heard of this problem before, but I did restrain myself from rehashing any arguments already put forth. And yes, I did read the Wiki link, and a bunch of the other links.

Which is fine, and I admire your desire to understand these things better. The problem is that mathematics is the one field where there are, to some extent, universal truths rather than just statements which contain a degree of uncertainty or subjectivity, and so at the end of the day these threads tend to devolve into the people who understand the detailed maths pointing out the flaws in arguments, and the people who refuse to change their mind sticking their fingers into their ears, and anyone who came in for "a little convo" either gets converted one way or the other, or driven off. It's like ... well, it's like pretty much any of the topics labelled as "done to death".

If there's something in an existing explanation that you don't understand, or don't think is fully justifiable, though, feel free to point it out. I'm sure there's still someone willing to either explain it further or make it more rigorous. The "1/3 x 3 = 0.333... x 3" is not one of the stronger arguments, but the ones that build up from a definition of a decimal are generally pretty good.
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### Re: A very interesting Mathematical Paradox

here's a thought, 1 = 0.111... in binary, or in general 1 = 0.nnn... in base n+1

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### Re: A very interesting Mathematical Paradox

Well, though normally something like this should have been in the math forum.

But since they have viewtopic.php?f=17&t=14321 it is actually useless to post this thread anywhere...

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### Re: A very interesting Mathematical Paradox

UrielZyx wrote:Well, though normally something like this should have been in the math forum.

But since they have viewtopic.php?f=17&t=14321 it is actually useless to post this thread anywhere...

If you look at the dates, this precedes that notice, and even the creation of the maths forum.
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### Re: A very interesting Mathematical Paradox

Ugh. I made this thread on another site, and got 500 replies in a day. Each of them could have been blown down with about three seconds of thought, and were all indicitive of the fact that some people just don't get this concept.

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### Re: A very interesting Mathematical Paradox

Thank you, ConMan, for being gentle with your convictions and arguments. I appreciate the treatment. And that comic is hilarious. It's a humorous way to express my frustration with that aspect of the Math and reality issue. lol.

ConMan wrote:...universal truths rather than just statements which contain a degree of uncertainty or subjectivity...

I think people believe in the infallibility of numbers too much. It seems like so many things are approximations. We trust computers (number machines), but just as they cannot be truly random they cannot be truly accurate. The real world is the most accurate thing out there and who the heck knows what's really going on there.

P.S. You seem uber smart, so I concede to all arguments in your presence.

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### Re:

wisnij wrote:It doesn't work that way. The string of 9s is infinitely long; there is no last digit. Every digit 9 in x gets matched up with a digit 9 in 10x, and they all cancel just as described initially.

That is not inherently necessary. There can be a last digit in an "infinitely long" string of digits - the string just has to have a successor order type. Real numbers, however, have a decimal expansion of order type ω, for which there is indeed no last digit, nor any digit preceded by infinitely many digits (which is still possible in a string of digits with no last digit, and sufficient for infinitesimals!) as all digits in a real number are preceded by only finitely many other digits and have a finite place.

Now of course, we have to invent a new number system where digits can have any ordinal place value.

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### Re: A very interesting Mathematical Paradox

So, how 'bout we take this to the "we already know about the .99999... = 1" thread on the mathematics forum?

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### Re: A very interesting Mathematical Paradox

...999 + 1 = 0 looks a lot like certain other expressions.

255+1=0 in 8-bit representation
65535+1=0 in 16-bit representation.

...999+1=0. The real numbers loop around. We are living inside a computer.

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### Re: A very interesting Mathematical Paradox

xkcdfan wrote:...999 + 1 = 0 looks a lot like certain other expressions.

255+1=0 in 8-bit representation
65535+1=0 in 16-bit representation.

...999+1=0. The real numbers loop around. We are living inside a computer.

... what? Something that we invent resembles other things that we've invented, therefore we know something about how the universe works?? really???

t1mm01994 wrote:So, how 'bout we take this to the "we already know about the .99999... = 1" thread on the mathematics forum?

I like that this is open. So, no, that's not happening.
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### Re: A very interesting Mathematical Paradox

jestingrabbit wrote:
xkcdfan wrote:...999 + 1 = 0 looks a lot like certain other expressions.

255+1=0 in 8-bit representation
65535+1=0 in 16-bit representation.

...999+1=0. The real numbers loop around. We are living inside a computer.

... what? Something that we invent resembles other things that we've invented, therefore we know something about how the universe works?? really???

Yes.

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### Re: A very interesting Mathematical Paradox

xkcdfan wrote:Yes.

I've gotta say that's pretty inane.
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### Re: A very interesting Mathematical Paradox

xkcdfan wrote:
jestingrabbit wrote:
xkcdfan wrote:...999 + 1 = 0 looks a lot like certain other expressions.

255+1=0 in 8-bit representation
65535+1=0 in 16-bit representation.

...999+1=0. The real numbers loop around. We are living inside a computer.

... what? Something that we invent resembles other things that we've invented, therefore we know something about how the universe works?? really???

Yes.
I do not think you understand him. You see, humans invented computers. The only reason they loop around like that is because humans make it that way. Numbers, however, are not made (in most cases) just by however humans want them to. Just because humans made computers work a certain way does not mean anything about anything else.

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### Re: A very interesting Mathematical Paradox

Mike_Bson wrote:Numbers, however, are not made (in most cases) just by however humans want them to.

No I meant that that's exactly how numbers are. They have some relationship to reality, but its the same sort of relationship that Newtonian mechanics has. Its a good approximation to the truth, but it is not the truth. We invented the real numbers to work in a particular way. The idea that there is some sense in ...999 is a purely human invention, not something that is true about the universe. The idea that a bunch of similarities that exist between things that humans have invented might have some bearing on the nature of the universe is inane.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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### Re: A very interesting Mathematical Paradox

jestingrabbit wrote:
Mike_Bson wrote:Numbers, however, are not made (in most cases) just by however humans want them to.

No I meant that that's exactly how numbers are. They have some relationship to reality, but its the same sort of relationship that Newtonian mechanics has. Its a good approximation to the truth, but it is not the truth. We invented the real numbers to work in a particular way. The idea that there is some sense in ...999 is a purely human invention, not something that is true about the universe. The idea that a bunch of similarities that exist between things that humans have invented might have some bearing on the nature of the universe is inane.

I get what you're saying, now.

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### Re: A very interesting Mathematical Paradox

You're assuming that nobody invented numbers.

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### Re: A very interesting Mathematical Paradox

xkcdfan wrote:You're assuming that nobody invented numbers.

No, nobody's assuming that. In fact, they're using that as the main point against your nonsense.

That is: Why should the properties of something we invented (specifically, a number (...999) in a number system that was invented in this thread by someone inspired by a quirk in a also-human-invented representation of a different number system (the reals) that was also human-invented but is an attempt to model the real world, which is an imperfect model but very useful) have any relation to a real world it's 4 degrees of separation away from?

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AWA
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### Re: A very interesting Mathematical Paradox

I didn't read the entire thread, so I don't know if this was addressed already, but my main beef with the whole 0.999... = 1 thing is this:

If one were to plot the point (1,0) on a graph, one would expect that it would be identical to the point ((0.999...),0).

However:

If you construct the point that is 0.999... units from the origin by its decimal expansion like so...

1. Plot point (.9,0), that is, 9/10 of the way from 0 to 1.
2. Plot point (.99,0), that is, 9/10 of the way from .9 to 1.
3. Plot point (.999,0), that is, 9/10 of the way from .99 to 1.

...then, even though you will get exponentially closer to (1,0), it will not actually equal the point (1,0). I understand that there is no "last point" to plot; I'm saying that no matter how many points are plotted, up to and including an infinite number of points, the function (term used for simplicity) of plotting the point 9/10 of the way to the destination will never yield (1,0).

I get the algebraic proofs (though to those I conjecture that the repeating decimal 0.999... only approaches 1, as 0.333... approaches 1/3 and 0.666... approaches 2/3). I cannot, however, compromise these with this simple geometric counter-proof.

Qaanol
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### Re: A very interesting Mathematical Paradox

AWA, are you familiar with the Archimedean property of the real numbers? It states that for any positive real number x, there is a positive integer n such that [imath]0<\frac{1}{n}<x[/imath]. This is a non-negotiable consequence of the definition of real numbers. I want you to think about what it means.
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phlip
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### Re: A very interesting Mathematical Paradox

Plot the point (17,0) on your graph
Plot the point (83.2,0) on your graph
Plot the point (-45,0) on your graph

Depending on your sequence of random numbers, you could very well avoid ever landing exactly on 1. Does this prove anything? Of course not, because none of those numbers are 0.999....

And neither are any of yours. Each of your numbers is closer to 0.999... than the one before it, but none of them is actually 0.999....

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jestingrabbit
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### Re: A very interesting Mathematical Paradox

AWA wrote:I get the algebraic proofs (though to those I conjecture that the repeating decimal 0.999... only approaches 1, as 0.333... approaches 1/3 and 0.666... approaches 2/3). I cannot, however, compromise these with this simple geometric counter-proof.

I think this is where your problem is. You talk about 0.999... approaching 1. 0.999... doesn't approach anything. Its a constant. The sequence 0.9, 0.99, 0.999, 0.9999, etc approaches 1, and 0.999... is what that sequence approaches, not the sequence doing the approaching.

Equally, 0.333... is 1/3, 0.666... is 2/3 and pi is 3.141... , where we take "..." to be the correct infinite sequence of digits. Sure none of those can be represented with a finite number of digits, but the real numbers aren't just the numbers with finite decimal expansions, but the numbers with infinite decimal expansions as well.
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AWA
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### Re: A very interesting Mathematical Paradox

But, I'm intending to construct the repeating decimal. I'm just stopping along the way. If we can suppose that We're trying to reach (0.4368,0), then we can construct that by first plotting (0.4,0), then (0.43,0), then (0.436,0), and finally (0.4368,0). Here, I'm trying to construct (1,0) by infinitely iterating a process of plotting the next decimal. Again, I understand that there is no "last" decimal; my point is that it doesn't matter. it will never reach (1,0).

phlip
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### Re: A very interesting Mathematical Paradox

So... you're trying to "construct" a number by looking at other, irrelevant numbers?

Of course 0.4 ≠ 0.43 ≠ 0.436 ≠ 0.4368. Is there anything at all surprising about that?
Last edited by phlip on Mon Aug 23, 2010 4:37 am UTC, edited 1 time in total.

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