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Thread: 1 = 2 'proof' ... what's wrong?

  1. #1 1 = 2 'proof' ... what's wrong? 
    Forum Ph.D. william's Avatar
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    Okay, here's another one.

    Theorem: 1 = 2
    'Proof:'
    Let a = b = 1
    Then a = b
    a^2 = a*b
    a^2 - b^2 = a*b - b^2
    (a + b)(a - b) = b(a - b)
    a + b = b
    1 + 1 = 1
    therefore, 2 = 1
    Obviously, this is just as much bunk as the last one I gave.
    So what's the mathematical error?

    Cheers,
    william

    Edit:
    Vroomfondel PM'd me the correct answer. I suppose he didn't want to post it and spoil the fun so soon.
    Thanks vroom, and good job!


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  3. #2  
    Forum Professor river_rat's Avatar
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    Lol - that ones easy

    For a more interesting example try these two:

    Proof that 2 = 1 :

    Log 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ....
    = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
    = ((1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...)) - 2 (1/2 + 1/4 + 1/6 + ...)
    = (1 + 1/2 + 1/3 + 1/4 + ...) - (1 + 1/2 + 1/3 + 1/4)
    = 0 = Log 1

    So Log 2 = Log 1 => 2 = 1 as natural logarithms are 1-1 for real numbers.

    QED

    AND

    Proof that pi = e

    Lemma 1 : Any positive integer is equal to any positive integer that does not succeed it

    Proof (by induction) :

    The statement is obviously true for n = 1 as 1 = 1 and there is no smaller positive integer other that 1.

    So suppose true for all 1 < n <= k and consider the integer k. Then by induction hypothesis k = k-1 as k-1 is a positive integer
    that does not succeed k and thus k = k + 1 and thus the theorem is true for all positive integers.

    Proof :

    2 <= e <= 3 and thus 3 = 2 (by lemma) and so 3 <= e <= 3 and so e = 3. Similarly we have that pi = 3 and thus pi = e

    QED


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  4. #3  
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    Quote Originally Posted by river_rat
    Lol - that ones easy

    For a more interesting example try these two:

    Proof that 2 = 1 :

    Log 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ....
    = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
    = ((1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...)) - 2 (1/2 + 1/4 + 1/6 + ...)
    = (1 + 1/2 + 1/3 + 1/4 + ...) - (1 + 1/2 + 1/3 + 1/4)
    = 0 = Log 1

    So Log 2 = Log 1 => 2 = 1 as natural logarithms are 1-1 for real numbers.

    QED

    AND

    Proof that pi = e

    Lemma 1 : Any positive integer is equal to any positive integer that does not succeed it

    Proof (by induction) :

    The statement is obviously true for n = 1 as 1 = 1 and there is no smaller positive integer other that 1.

    So suppose true for all 1 < n <= k and consider the integer k. Then by induction hypothesis k = k-1 as k-1 is a positive integer
    that does not succeed k and thus k = k + 1 and thus the theorem is true for all positive integers.

    Proof :

    2 <= e <= 3 and thus 3 = 2 (by lemma) and so 3 <= e <= 3 and so e = 3. Similarly we have that pi = 3 and thus pi = e

    QED

    Curious, I always thought the log of 2 was 0.30103 +/- a very small part.
    Your series suggests it will be between 0.5 and 1.0

    Are you by any chance a tax collector?
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  5. #4  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by billco
    Curious, I always thought the log of 2 was 0.30103 +/- a very small part.
    Your series suggests it will be between 0.5 and 1.0

    Are you by any chance a tax collector?
    No Log(2) = 0.693147181..... (Are you using base e?)
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  6. #5  
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    Quote Originally Posted by river_rat
    Quote Originally Posted by billco
    Curious, I always thought the log of 2 was 0.30103 +/- a very small part.
    Your series suggests it will be between 0.5 and 1.0

    Are you by any chance a tax collector?
    No Log(2) = 0.693147181..... (Are you using base e?)

    Perhaps you could tell me when it changed?

    Log(2) = 0.30103 +/- a very small part

    ln(2) = 0.693147 +/- a very small part.
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  7. #6  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by billco
    Perhaps you could tell me when it changed?

    Log(2) = 0.30103 +/- a very small part

    ln(2) = 0.693147 +/- a very small part.
    Ive yet to meet a self-respecting mathematician who uses the ln notation, all logs in mathematics are to the natural base! In fact paul halmos heaped contempt on the very idea
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  8. #7  
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    Quote Originally Posted by river_rat
    Quote Originally Posted by billco
    Perhaps you could tell me when it changed?

    Log(2) = 0.30103 +/- a very small part

    ln(2) = 0.693147 +/- a very small part.
    Ive yet to meet a self-respecting mathematician who uses the ln notation, all logs in mathematics are to the natural base! In fact paul halmos heaped contempt on the very idea
    Well I'm just a humble retired design engineer My part of the space shuttle system is still used (unmodified!) - my contribution to the Development of the PC at (Boca Raton) has never been superseded
    what the phuck do I know about mathematics.


    All three of my Scientific calculators clearly distinguish 'ln' from 'log'

    And as for your ref to Paul Halmos,

    "Those who quote others have not the brains for wit themselves" Billco.
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  9. #8  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by billco
    Well I'm just a humble retired design engineer My part of the space shuttle system is still used (unmodified!) - my contribution to the Development of the PC at (Boca Raton) has never been superseded
    what the phuck do I know about mathematics.
    Great so your a good engineer - but mathematical notation and engineering notation do not gell well. When a mathematicians writes log, they assume the natural base as thats what you use to do analysis (or else Log_2 if you are working in computer science - though i would have assumed that someone has knowledgeable as yourslef would know that log in C++ returns the natural log), the context should have been clear from the fact that a taylor series had been used.

    Anyway, the series expansion is correct - now spot the error.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  10. #9  
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    Quote Originally Posted by river_rat
    Quote Originally Posted by billco
    Well I'm just a humble retired design engineer My part of the space shuttle system is still used (unmodified!) - my contribution to the Development of the PC at (Boca Raton) has never been superseded
    what the phuck do I know about mathematics.
    Great so your a good engineer - but mathematical notation and engineering notation do not gell well. When a mathematicians writes log, they assume the natural base as thats what you use to do analysis (or else Log_2 if you are working in computer science - though i would have assumed that someone has knowledgeable as yourslef would know that log in C++ returns the natural log), the context should have been clear from the fact that a taylor series had been used.

    Anyway, the series expansion is correct - now spot the error.
    Good engineers assume nothing, as ASS-U-ME as you can clearly see makes an ASS out of U and ME. Engineers check and double check and sometimes check 10,20 or even 100 times and then again, such that when the mathematicians walk upon the bridges we build their feet do not get wet.

    As for the mistake it is simply your opening statement [proof 2 =1]

    That is a mistake.


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  11. #10  
    Forum Isotope Zelos's Avatar
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    Ive yet to meet a self-respecting mathematician who uses the ln notation, all logs in mathematics are to the natural base!
    ln is base e, and log is base 10
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

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  12. #11  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by Zelos
    ln is base e, and log is base 10
    Which is fine if you are doing pre-calculus, but im yet to meet a mathematician who uses ln when talking about the natural logarithm (and ive met a few). The only logarithm generally used in analysis is the natural logarithm and it is denoted by Log(x).

    If you dont believe me try wikipedia or wolfram, or any analysis text book.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  13. #12  
    Forum Professor river_rat's Avatar
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    Quote Originally Posted by billco

    As for the mistake it is simply your opening statement [proof 2 =1]

    That is a mistake.
    Lol - anyone got a more serious answer?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  14. #13  
    Forum Ph.D. william's Avatar
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    Hi river_rat,
    Lol - that ones easy

    For a more interesting example try these two:

    Proof that 2 = 1 :

    Log 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ....
    = (1 + 1/3 + 1/5 + ...) - (1/2 + 1/4 + 1/6 + ...)
    = ((1 + 1/3 + 1/5 + ...) + (1/2 + 1/4 + 1/6 + ...)) - 2 (1/2 + 1/4 + 1/6 + ...)
    = (1 + 1/2 + 1/3 + 1/4 + ...) - (1 + 1/2 + 1/3 + 1/4)
    = 0 = Log 1

    So Log 2 = Log 1 => 2 = 1 as natural logarithms are 1-1 for real numbers.

    QED

    AND

    Proof that pi = e

    Lemma 1 : Any positive integer is equal to any positive integer that does not succeed it

    Proof (by induction) :

    The statement is obviously true for n = 1 as 1 = 1 and there is no smaller positive integer other that 1.

    So suppose true for all 1 < n <= k and consider the integer k. Then by induction hypothesis k = k-1 as k-1 is a positive integer
    that does not succeed k and thus k = k + 1 and thus the theorem is true for all positive integers.

    Proof :

    2 <= e <= 3 and thus 3 = 2 (by lemma) and so 3 <= e <= 3 and so e = 3. Similarly we have that pi = 3 and thus pi = e

    QED
    Well, these two are also trivial (depending on the level of mathematical sophistication one has). I shall try to be vague enough to where you understand but not to spoil the fun for anyone else.

    By the way, I like these 'proofs' of yours.



    Okay, the first fails because the series for ln(2) is conditionally convergent. The harmonic series (which you end up with - twice) is divergent.

    The second fails (and I'm not sure how to state this without 'spoiling the fun') due to an improper use of proof by induction. In this case, it is more appropriate to start with n = 2 instead of n = 1 because the lemma doesn't apply for n = 1 (i.e., there are no smaller positive integers than 1). I could try to be more sophisticated by couching this in a "...there exists a homomorphism between Z+ and Z..." yada yada but the above critique will suffice.

    Keep 'em comin'

    By the way, I use ln for base 'e' and log for base '10'. I think it's a matter of preference (unless I'm not self-respecting... ). I've seen it both ways. And I suppose I should add that I'm not a mathematician - but I am a physicist.

    Also, keep in mind the audience to whom you are conversing. From what I gather, we have a lot of high school students, engineers, everyday common folk just interested in science, and a smattering of scientists (professionals). The point here is to just have fun.

    Cheers,
    william
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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  15. #14  
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    Quote Originally Posted by billco
    All three of my Scientific calculators clearly distinguish 'ln' from 'log'
    "In our calculator we trust" -every engineer I've met

    In the vast majority of mathematics, texts, journal articles, lectures, etc. "log" will denote the logarithm to base "e". This is the most common and usefull log that comes up for mathematicians, so it has a good hold on this notation in maths. "ln" doesn't generally appear. The main exception is your calc texts aimed at science and engineering students, which will agree with your trusted calculators (which weren't made with mathematicians in mind).

    A capital "L" on the "Log" is rare, most common in my experience in complex analysis texts to denote a specific branch of the logarithm they commonly use throughout the text. They'll use "log" if they want to leave it unspecified.

    Notation isn't standard, get used to it. I've seen many lectures/articles where a subscript on the log denotes iterations (of course they explicitely explained this) i.e. log_3(x)=log(log(log(x))), because it was convenient to have a notation for such a thing, and the only base of log that will be encountered was 'e', (or the base just didn't matter, this is often enough the case).

    The moral, if you see "log" in a strange location and the base does matter, take a moment to figure out the context and if this determines the base. In the OP, it should be clear given this recognizable series (c'mon, you've all seen the series for log(1-x), even the engineers). If the base doesn't matter, just pretend it's your favorite if it makes you more comfortable.
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  16. #15  
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    Quote Originally Posted by shmoe
    Quote Originally Posted by billco
    All three of my Scientific calculators clearly distinguish 'ln' from 'log'
    "In our calculator we trust" -every engineer I've met

    In the vast majority of mathematics, texts, journal articles, lectures, etc. "log" will denote the logarithm to base "e". This is the most common and usefull log that comes up for mathematicians, so it has a good hold on this notation in maths. "ln" doesn't generally appear. The main exception is your calc texts aimed at science and engineering students, which will agree with your trusted calculators (which weren't made with mathematicians in mind).

    A capital "L" on the "Log" is rare, most common in my experience in complex analysis texts to denote a specific branch of the logarithm they commonly use throughout the text. They'll use "log" if they want to leave it unspecified.

    Notation isn't standard, get used to it. I've seen many lectures/articles where a subscript on the log denotes iterations (of course they explicitely explained this) i.e. log_3(x)=log(log(log(x))), because it was convenient to have a notation for such a thing, and the only base of log that will be encountered was 'e', (or the base just didn't matter, this is often enough the case).

    The moral, if you see "log" in a strange location and the base does matter, take a moment to figure out the context and if this determines the base. In the OP, it should be clear given this recognizable series (c'mon, you've all seen the series for log(1-x), even the engineers). If the base doesn't matter, just pretend it's your favorite if it makes you more comfortable.

    I guess my calculators were designed by engineers.....
    Either way if you ar talking log I'd like to be given the base. If I said 15 appeared over the horizon you'd ask what of? - If I said ask any Engineer - you'd get a bit annoyed. Do NOT assume everyone
    knows what you are talking about. A good engineer will NEVER assume
    he knows the units, he will always check!

    That's why Engineers make bridges, mathematicians make mistakes.
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  17. #16  
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    Quote Originally Posted by billco
    I guess my calculators were designed by engineers.....
    By or for, sure probably. and accountants.

    Quote Originally Posted by billco
    Either way if you ar talking log I'd like to be given the base. If I said 15 appeared over the horizon you'd ask what of? - If I said ask any Engineer - you'd get a bit annoyed. Do NOT assume everyone
    knows what you are talking about. A good engineer will NEVER assume
    he knows the units, he will always check!
    You were the one who assumed the base was 10...and were mistaken, so what is your point?

    Quote Originally Posted by billco
    That's why Engineers make bridges, mathematicians make mistakes.
    Since you appear to have complete ignorance of mathematicians, I won't take this totally unfounded statement personally. Are you trying to start some childish "engineers are better than mathematicians" fight?
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  18. #17  
    Forum Professor river_rat's Avatar
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    Hi william

    I must admit they are not my proofs (i picked them up browsing a while back, and really liked the induction one as at first glance the induction seems correct) but they really are cute

    Ill have to dig for a few more, do you have any lying around?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  19. #18  
    Forum Ph.D. william's Avatar
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    Hi river_rat (double-r),
    I have a couple more - one easy, one not so easy (again, depending on the level of math one is accustomed to). I may post them at a later time.

    The thing I like about these 'proofs' is that we are always told we can't do certain things in math - or that some things are 'undefined.' These 'proofs' demonstrate why these things are undefined - and why we can't do them.

    Cheers,
    william
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
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  20. #19  
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    Quote Originally Posted by shmoe
    Quote Originally Posted by billco
    I guess my calculators were designed by engineers.....
    By or for, sure probably. and accountants.

    Quote Originally Posted by billco
    Either way if you ar talking log I'd like to be given the base. If I said 15 appeared over the horizon you'd ask what of? - If I said ask any Engineer - you'd get a bit annoyed. Do NOT assume everyone
    knows what you are talking about. A good engineer will NEVER assume
    he knows the units, he will always check!
    You were the one who assumed the base was 10...and were mistaken, so what is your point?

    Quote Originally Posted by billco
    That's why Engineers make bridges, mathematicians make mistakes.
    Since you appear to have complete ignorance of mathematicians, I won't take this totally unfounded statement personally. Are you trying to start some childish "engineers are better than mathematicians" fight?

    If it get's it off your chest you may hurl insults as you please, "sticks and stones may break my bones but names, they cannot hurt me!"
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  21. #20 Back to the original proof 
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    William,
    Back to the original "proof". I see which step is wrong but I can't figure out why? The math seems ok even though it obviously isn't. Could you explain what's wrong for us "C" students. I have never built a bridge, nor created a $30,000 o-ring for a space shuttle so I'm not as good at this stuff as most of you. I still enjoy trying though. Thanks for the fun.

    Cheese
    "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. " Albert Einstein

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  22. #21  
    Forum Ph.D. william's Avatar
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    Hi Cheeseman,
    I just sent you a personal message with your answer. I don't want to answer this one myself on this thread.

    Cheers,
    william
    "... the polhode rolls without slipping on the herpolhode lying in the invariable plane."
    ~Footnote in Goldstein's Mechanics, 3rd ed. p. 202
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  23. #22 Re: Back to the original proof 
    Forum Sophomore Absane's Avatar
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    Quote Originally Posted by Cheeseman
    William,
    Back to the original "proof". I see which step is wrong but I can't figure out why? The math seems ok even though it obviously isn't. Could you explain what's wrong for us "C" students. I have never built a bridge, nor created a $30,000 o-ring for a space shuttle so I'm not as good at this stuff as most of you. I still enjoy trying though. Thanks for the fun.

    Cheese
    Well the original assumption is that a = b.

    Or, an equivalent way to write this is a - b = 0, correct?

    Well, in one of the steps you have this: (a + b)(a - b) = b(a - b)

    Well, in the next step you divide both sides by (a-b) and get: a+b = b.

    However, dividing by a-b is dividing by zero since a-b = 0.

    And we all know that division by zero produces a lot of weird results and just simply makes no sense.
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  24. #23 Confirmed: I'm a moron 
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    Thanks guys for the help. When you see it is so simple but up until that point I was dumbfounded. Thanks again.
    "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. " Albert Einstein

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