1. Zugzwang
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    12 Nov '14 19:562 edits
    Recently, some people here were complaining that the proposition that
    1/2 + 1/4 + 1/8 + ... = 1 was a 'paradox' (or simply too confusing).

    After some mathematical lectures at Harvard, Benjamin Peirce (1809-1880) reportedly said:
    "Gentlemen, this is surely true, it is absolutely paradoxical. We cannot
    understand it, and we don't know what it means. But we have proved it,
    and therefore we know it must be the truth." He was referring to:
    i^i = e^(-pi/2) ~ 0.2078... (which he called a 'mysterious formula' ).

    By the way, speaking of his usage of 'gentlemen', no women evidently attended
    Benjamin Peirce's lectures. He, a devout Christian, approved of slavery.
    Benjamin Peirce was very much an American man of the (early) 19th century.

    "Mathematics is the science that draws necessary conclusions."
    --Benjamin Peirce (1866, in _Linear Associative Algebra_)
  2. Subscribersonhouse
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    13 Nov '14 17:37
    Originally posted by Duchess64
    Recently, some people here were complaining that the proposition that
    1/2 + 1/4 + 1/8 + ... = 1 was a 'paradox' (or simply too confusing).

    After some mathematical lectures at Harvard, Benjamin Peirce (1809-1880) reportedly said:
    "Gentlemen, this is surely true, it is absolutely paradoxical. We cannot
    understand it, and we don't know what it means. ...[text shortened]... e that draws necessary conclusions."
    --Benjamin Peirce (1866, in _Linear Associative Algebra_)
    the paradox being i squared we define as being minus one? i being the square root of minus one.
  3. Standard memberSoothfast
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    13 Nov '14 18:24
    Originally posted by sonhouse
    the paradox being i squared we define as being minus one? i being the square root of minus one.
    I'm not actually sure why i^i=e^(-pi/2) was considered paradoxical by Peirce. Perhaps because we see here an imaginary exponentiated by an imaginary, and the result is a real exponentiated by a real. "Paradoxical" is being used here to mean "unexpected" or "strange," I think, as it often is in everyday language.
  4. Cape Town
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    13 Nov '14 18:36
    Originally posted by Duchess64
    Recently, some people here were complaining that the proposition that
    1/2 + 1/4 + 1/8 + ... = 1 was a 'paradox' (or simply too confusing).
    Actually, no, nobody thought that was a paradox. What seems paradoxical is the concept that an object can traverse the infinite sequence 1/2 + 1/4 + 1/8 + ... and yet later proceed to further locations. The paradox is that it appears that the object has got to infinity and beyond.
  5. Standard memberDeepThought
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    13 Nov '14 18:42
    Originally posted by Soothfast
    I'm not actually sure why i^i=e^(-pi/2) was considered paradoxical by Peirce. Perhaps because we see here an imaginary exponentiated by an imaginary, and the result is a real exponentiated by a real. "Paradoxical" is being used here to mean "unexpected" or "strange," I think, as it often is in everyday language.
    It's possible to represent i as a two by two matrix, so in that representation the formula i^i has a matrix raised to the power of a matrix, which probably is paradoxical.
  6. Zugzwang
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    13 Nov '14 22:461 edit
    Originally posted by twhitehead
    Actually, no, nobody thought that was a paradox. What seems paradoxical is the concept that an object can traverse the infinite sequence 1/2 + 1/4 + 1/8 + ... and yet later proceed to further locations. The paradox is that it appears that the object has got to infinity and beyond.
    Does Twhitehead claim to read everyone else's mind with certainty?

    In fact, Shavixmir created a thread, 'Something drops half a distance', in
    which his original post (asking for a 'mathematical explanation' ) led to
    several writers referring to the problem as a 'paradox'. For examples,

    "This is known as Zeno's Dichotomy Paradox."
    --PatNovak (22 October 2014)

    "It is, like the quantification of space, irrelevant to this paradox..."
    --KazetNagorra (23 October 2014)

    "It is perfectly reasonable to defeat a paradox by defeating one of its underlying assumptions."
    --PatNovak (23 October 2014)

    "I looked up the word paradox to see if this really is a paradox or not.
    Apparently it is."
    --Lemon Lime (6 November 2014)

    With commendable honesty, Shavixmir and GreatKingRat wrote that they
    struggled to comprehend the various explanations (mathematical, physical, or
    philosophical) being offered about this 'paradox'. I suspect that other readers
    also were confused. I don't claim to know exactly why they were confused.

    By the way, if Twhitehead were better at reading my mind, then he might be
    able to solve some recreational mathematical problems that I solved in childhood.
  7. Standard memberSoothfast
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    14 Nov '14 02:49
    Originally posted by DeepThought
    It's possible to represent i as a two by two matrix, so in that representation the formula i^i has a matrix raised to the power of a matrix, which probably is paradoxical.
    Yes, though it's much more common to formally regard complex numbers as ordered pairs: a+ib = (a,b). A plane is the natural habitat of the complex numbers, as opposed to presenting them as some subspace of a 4-dimensional space. Then the real numbers are simply identified with ordered pairs having second coordinate equal to zero: (a,0) = a. I'm sure in Peirce's time this was being done. I'm not sure whether two-by-two matrix representations were being entertained back then. It's possible. Looking around, it seems that Arthur Cayley may have conceived of the idea in 1858.
  8. Cape Town
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    14 Nov '14 05:24
    Originally posted by Duchess64
    Does Twhitehead claim to read everyone else's mind with certainty?.
    No.
    I admit that you may have mistakenly thought the sum you gave was a paradox.
    But Zeno's Paradox is about traversing the sequence not obtaining the sum.
    But then you seem to be incapable of admitting when you are wrong.
  9. Zugzwang
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    14 Nov '14 06:193 edits
    Originally posted by twhitehead
    No.
    I admit that you may have mistakenly thought the sum you gave was a paradox.
    But Zeno's Paradox is about traversing the sequence not obtaining the sum.
    But then you seem to be incapable of admitting when you are wrong.
    Twhitehead's an extremely arrogant and dishonest troll who already has quite
    a record of misrepresenting me and routinely distorting the facts in order
    to make more personal attacks against me. I expect Twhitehead to keep
    lying--such is his base character--in order to keep personally attacking me.

    Although he claims to hold a university degree in mathematics, evidently
    Twhitehead's quite weak in mathematics and cannot make any meaningful
    independent contribution (looking up stuff on YouTube does not qualify)
    in a mathematical discussion. After apparently implicitly accusing me of
    making up my mathematical background, Twhitehead was unable to solve
    even a few recreational mathematical problems that I solved as a young
    student. His inability to do such simple mathematics did not prevent
    Twhitehead, of course, from his characteristic sneering at my intelligence.

    Twhitehead warrants no response beyond absolute disdain.
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    14 Nov '14 07:591 edit
    Slightly confusing question:

    Tim is twice the age of what Sue was when Sue was 8 years younger than the age Tim is now.
    How old is Tim?

    ( just an attempt to lighten-up this atmosphere )
  11. Cape Town
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    14 Nov '14 08:49
    Originally posted by Duchess64
    Twhitehead warrants no response beyond absolute disdain.
    I hope you at least felt a little embarrassed writing all that. One day, I hope you go over these threads in a calmer mindset and realize your error.
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    14 Nov '14 12:23
    Originally posted by twhitehead
    Actually, no, nobody thought that was a paradox. What seems paradoxical is the concept that an object can traverse the infinite sequence 1/2 + 1/4 + 1/8 + ... and yet later proceed to further locations. The paradox is that it appears that the object has got to infinity and beyond.
    The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of this website where that was stated (maybe even by you), but it was some time ago and don’t remember which thread it was.

    My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
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    14 Nov '14 13:51
    Originally posted by Great King Rat
    The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of thi ...[text shortened]...

    My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
    You say "infinitely close to 1[/i]" and not exact 1, right?

    As this is mathematics, then this would be provable. Please give us the proof, along with the definitions needed for the proof to be stringent.
  14. Standard memberSoothfast
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    14 Nov '14 15:09
    Originally posted by humy
    Slightly confusing question:

    Tim is twice the age of what Sue was when Sue was 8 years younger than the age Tim is now.
    How old is Tim?

    ( just an attempt to lighten-up this atmosphere )
    Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

    -Gustave Flaubert
  15. Standard memberSoothfast
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    14 Nov '14 15:161 edit
    Originally posted by Great King Rat
    The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1. And as far as I know it is considered wrong to treat infinity as a number. I also remember reading some thread(s) in the spirituality forum of thi ...[text shortened]...

    My answer to what is the sum of 1/2 + 1/4 + 1/8 + ... would be infinitely close to 1.
    Here's some arithmetic sleight of hand you see in "survey of mathematics" texts: what is the difference between 1 and the nonterminating decimal 0.999999…?

    Let N = 0.999999…

    Then 10N = 9.999999…

    Now, 9N = 10N - N = 9.999999… - 0.999999… = 9.000000…

    Hence 9N = 9.

    So N = 1.

    Therefore 0.999999… = 1.

    Tah-dah!

    (The argument is somewhat informal, but it is sound.)
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