1=0.999... infinities and box of chocolates..Phliosophy of Math...

Discussion in 'General Philosophy' started by Quantum Quack, Nov 2, 2013.

  1. James R Just this guy, you know? Staff Member

    Motor Daddy:

    You forgot to reply to some of my replies to you.

    In particular, this one and this one.
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  3. James R Just this guy, you know? Staff Member


    What does the "first number" mean, anyway? You need to specify that you're talking about counting numbers if you want to say "1 is the first number".

    I mean, if I list three phone numbers: 555-1234, 555-7632 and 555-2356 then the first number is 555-1234.

    On another issue, you're still getting it wrong. If 0.999... = 1 (which it does), then they are two different representations of the same number.

    It makes no sense, then, to ask if 0.999... comes before or after 1. They are the same number.

    Get it?
    Edit to add:

    And on the other example you were confused about:

    0.25 + 0.25 + 0.25 + 0.25 = 0.999.... = 1

    because this is equivalent to

    0.24999.... + 0.24999... + 0.24999... + 0.24999.... = 0.999....

    Let me give you a few more examples:

    99.9999... = 100
    0.741239999.... = 0.74124
    65.19999.... = 65.2

    See the pattern?

    The fact that we can write the same number as 1 or as 0.99... is just an artifact of our base-10 number system. I have shown you that similar things happen in any base.
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  5. hansda Valued Senior Member

    So, essentially you are saying that:

    \( \lim_{n \to \infty} S_n = S \) where \( S = 1 \) and \( \lim_{n \to \infty} T_n = 0 \).

    Am i right, this time?
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  7. James R Just this guy, you know? Staff Member

    Yes, that's right hansda.
  8. Motor Daddy Valued Senior Member

    So a dollar isn't 4 .25 pieces, it's 4 .24999... pieces. No wonder I'm all screwed up.

    Yes. The left sides all have ... at the end, which means, to be completed. Just wondering how you concluded the right side when the left side is not complete. In other words, they all say, "infinite=finite." Got it, James. Thanks for pointing that out. I appreciate it!

    So .9=1 because .1+.1+.1+.1+.1+.1+.1+.1+.1=.9, so .9 is 100%, not 90% as I always used to think.
  9. hansda Valued Senior Member

    If you say \( \lim_{n \to \infty} T_n = 0 \) but \( T_n \ne 0 \) for any value of \( n \) because \( T_n \) is a term of a geometric series and so it can not be \( 0 \).

    \(T_n\) can only tend towards \(0\) but \(T_n\) can not be equal to \(0\).

    I think symbolically we can write this as: " \( T_n \to 0 \) as \( n \to \infty\)".
  10. James R Just this guy, you know? Staff Member

    Motor Daddy:

    0.24999... = 0.25

    They are the same number.

    I won't speculate on why you're screwed up.

    No. The dots mean only that there's an infinite string of 9s there.

    A number is not a process. It's a number. You don't construct numbers. You don't start building a number and then have your local council sign off on it once it's complete. A number just is.

    That's what the thread is all about.

    0.999... is an infinite string of nines after a decimal point, which can also be represented as the single digit "1".

    So, yes, in this case an infinite string represents the same number as the finite string.

    I'm glad you understand now.

    Do that again and I'll ban you for trolling. I'm serious.

    You know that nobody is arguing that 0.9 = 1. What is being argued is that 0.999... = 1.

    Clearly, 0.9 is different from 0.99, which is different from 0.999999, which is different from 0.999999999999, which is different from 0.999... Only the last of these is equal to 1.


    And you've still (conveniently) forgotten post #1041, above.

    I know you're trying to drop your silly pizza argument now, but you can't escape that easily. I won't have you forgetting what has gone before, like you forgot about our relativity box discussion. Age is no excuse. You must do better.
  11. Motor Daddy Valued Senior Member


    So are .9 and 1, they are 100%. The same. See?

    You'd ponder that infinitely, and then round it off to, "he's not."
  12. Baldeee Valued Senior Member

    From what I recall from Maths class, this is because we cannot actually reach infinity.
    But if we could, \(T_\infty\) = 0.
    If it doesn't = 0 then there must be another \( T_n_+_1\) that is half way between \(T_n\) and zero, and thus n would not yet have reached \(\infty\)
  13. hansda Valued Senior Member

    \(T_\infty \ne 0\), because \(T_\infty\) is still a term of the geometric series.
  14. Motor Daddy Valued Senior Member

    Yeah, and there is infinite distance in space, but I don't go around claiming that since distance never ends it equals 1, that is absolutely insane!

    HOW MANY 9's are after the point, James? When you answer that question in a finite way, then you've done something. To maintain a position that says that the two words "infinite" and "finite" are equal is just pure insanity! Nothing less!
  15. Baldeee Valued Senior Member

    If \(T_\infty \ne 0\), then let's say \(T_\infty = X\).
    According to the geometric series there must be an \(n\) that exists such that \(T_n < X\) but \(>0\)
    Thus \(n \ne \infty\)

    The only situation where this wouldn't be the case is where \(T_\infty = 0\)
  16. hansda Valued Senior Member


    I never said \(T_\infty = 0 \).
  17. James R Just this guy, you know? Staff Member


    You've posted a huge long post, yet still failed to answer the question I asked you. I remind you:

    1/2 + 1/4 + 1/8 + ... = 1. True or false?

    This is a mathematical statement. Either you think it is true, or you think it's false. There's no half-way house. So, which is it?

    Let's look at some of what you wrote. I really don't want to wade through all of it.

    I have not demanded anything of you. Here you are, making post after post in this thread, yet you seemingly don't have an actual point of view on the simple question above. All I see is waffle.

    Also, didn't anybody ever tell you that using ALL CAPITALS on the internet is considered to be SHOUTING? Do you want to look like some kind of deranged nut? Because that's the kind of person who mostly uses random capitals on the internet.

    I'm having trouble unpacking what you even mean by all that. All those CAPITALS for emphasis, and you still haven't communicated.

    Very often indeed, I'm sure. What's wrong with that?

    Not by me. Not at this stage.

    I wouldn't have thought that answering a simple question like the one above, about your own personal beliefs, would be as hard as it apparently is for you. Maybe you should just leave this thread alone.

    Does this mean that you're leaning towards saying the statement above is false, then? I really can't tell.

    Forget my arguments. How do you respond to post #915, above? That has put the nail in the coffin in this thread, as far as I'm concerned. What's your analysis of that?

    Nobody has asked you about reality and unreality. For my part, I simply want to know whether you agree with the truth of the above mathematical statement. It's a formal mathematical statement in a formal mathematical system. So, within that system, is it true or false?

    And, while we're at it, is it true that, in the same system, 0.999... = 1?

    One you've answered those questions, then maybe we can start getting all philosophical and start pondering whether this formal mathematics actually works in "real" world, whatever that is.

    Ok. So help me.

    Explain to me what is patently incomplete about my orthodoxy. Because it isn't patent to me at the present time.

    Do you include post #915 in that?

    You seem to think that a number is a process. Or do I have you wrong?

    Also, I'm puzzled as to what you think is "real" about a number such as 7 or 1 or pi. I'd say all numbers are abstractions. We can talk about the number 3, but it's an abstraction until we start talking about 3 sheep or 3 dollars or 3 tons of salt.

    It now sounds like you're saying you have no idea whether 0.999... = 1 or not. Is that correct?

    I think you've misunderstood Goedel. Of course you can use arguments within a formal system to prove statements made within that system. If you couldn't all mathematics would be useless. Goedel's theorems are about the completeness and consistency of formal systems.

    So, tell me how REALITY would work out whether 0.999... = 1 or not.

    I don't think Motor Daddy has pointed out anything of the kind to me.

    Maybe you can explain without all the CAPITALS and waffle.

    I thought it nicely demonstrated the point that 1 thing can be divided equally into 3 parts, which was an issue Motor Daddy disputed at the time. Now, he appears to be running away from that silly claim. But perhaps you'd like to take up his baton?

    Nobody's talking "real" yet. This thread is about the mathematical truth or falsity of 0.999... = 1, isn't it?

    This sounds (a) like you think that 1/2 + 1/4 + 1/8 + ... doesn't equal 1, and (b) like you think a number is a process.

    Do you want to address either of these points in more depth?
  18. James R Just this guy, you know? Staff Member

    Motor Daddy:

    Since your post was 4 minutes after mine, I'm going to give you the benefit of the doubt and assume you hadn't read my post.

    Next time, you're done for trolling. Just so you know.

    You're obviously confused. Look:

    0.9 + 0.09 + 0.009 + ... = 1

    See how the terms on the left hand side get smaller and smaller and smaller as you go to the right?

    Tell me: why do you think that sum will necessarily be infinite rather than finite? Or is this just another silly Motor Daddy gut feeling with nothing to back it up? Similarly, consider:

    1/2 + 1/4 + 1/8 + ... = 1

    This is the one that Undefined is struggling so hard with.

    Space, by the way, is not the same as a number. See my comments to Undefined, above, for example.

    Let's stick to the mathematics for now. We can sort out space later. You have to walk before you can run.

    This is looking an awful lot like trolling, MD.


    There is an infinite number of 9s after the point.
    I have never claimed that the words "finite" and "infinite" mean the same thing. If that's what you say you're getting from my posts then I can only conclude that (a) you're stupid; or (b) you're trolling; or (c) you're not really paying attention; or (d) one or more of the above.

    If you're confused about something, ask me a question. But don't you dare put words into my mouth again.
  19. Baldeee Valued Senior Member

    I know you didn't.
    I did.
    It is the conclusion of the part you accepted with "OK" plus my previous post.

    In the geometric progression there exists a term \(T_n\) such that \(T_n > T_n_+_1 > 0\)

    If \(T_\infty \ne 0\) then let \(T_\infty = X\)

    Now if \(X \ne 0\) then there must be a term \(T_n = X/2\) (if talking of the geometric progression of continually adding half the previous term etc).
    I.e. there must be a term \(T_n\) such that \(X > T_n > 0\)

    But if \(X = T_\infty\) then you're saying that \(T_\infty > T_n > 0\)

    So what is \(n\)?
    How can it be \(\infty + 1\)?

    So, as said previously, the only \(X\) such that \(T_\infty = X\) such that there is no \(T_\infty > T_n > 0\) is when \(X = 0\).
    i.e. when \(T_\infty = 0\).
  20. rpenner Fully Wired Valued Senior Member

    Actually because \(\infty\) is not a counting number; so, whatever \(T_{\infty}\) is supposed to mean, it is not part of the family of values \(T_n\).

    If \(\infty\) were a counting number then \(10^{\infty} \gt \infty^10 \gt \infty + 1 \gt \infty\). Actual number systems that use numbers larger than any finite number have infinite families of the infinitely large, but don't use the symbol \(\infty\) because that symbol already has too many different meanings.
    Specifically in the hyperreals where H is any number larger than all finite numbers, \(\lim_{n \to \infty} (1 - 10^{-n}) = \textrm{st}\left( 1 - 10^{-H} \right) = 1 - \textrm{st}\left( 10^{-H} \right) = 1 - 0 = 1\) because by definition \(\lim_{n \to \infty} A_n\) is equal to the "standard part" of the expression \(A_n\) as n is replaced with any infinitely large number so long as that "standard part" doesn't depend on which infinitely large number is being used.
  21. hansda Valued Senior Member

    What point you are trying to prove here?
  22. hansda Valued Senior Member

    May be you are right but do you think \(T_n\) being a valid member of a geometric series can ever be \(0\) for any value of \(n\)?
  23. rpenner Fully Wired Valued Senior Member

    Only in math systems where \(\infty + 1 = \infty\) like the extended real line. In such number systems \(0 = \frac{9}{\infty} = \frac{9}{10^{\infty}} = 9 \times 10^{-\infty}\) But it is still bending the rules since the family of \(T_n\) was only defined for n in the counting numbers. Since \(\infty\) is not a counting number, this \(T_{\infty}\) is still not part of the family being considered.

    So just in number systems where \(\infty + 1 = \infty\), we can correctly write \(\lim_{n\to \infty} S_n = S_{\infty} = S = 1\) while in the ordinary reals or hyperreals we can write \(\lim_{n\to \infty} S_n = S = 1\) and still be correct.

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