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Posers and Puzzles

Posers and Puzzles

  1. Standard member Nemesio
    Ursulakantor
    22 Aug '05 07:24
    I recently learned that:

    Pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...&c &c

    This seems unbelievably fascinating to me. First, because pi which is utterally
    irrational would have such a seemingly elegant mathematical defintion, and
    second that this was discovered at all.

    Does any mathematician want to talk about this defintion, how it came about,
    how it is applied?

    Nemesio
  2. 22 Aug '05 07:53
    Originally posted by Nemesio
    I recently learned that:

    Pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...&c &c

    This seems unbelievably fascinating to me. First, because pi which is utterally
    irrational would have such a seemingly elegant mathematical defintion, and
    second that this was discovered at all.

    Does any mathematician want to talk about this defintion, how it came about,
    how it is applied?

    Nemesio
    I am not a mathematician, but I have read a few things about it.

    The series was found by Leibniz as an approximation for pi. It is derived from the arctan(x) gregory series (James Gregory) for x=1 (which indeed equals pi/4).

    It looks great, BUT the series is coverging extremely slowly, hence not very practical. An accuracy of 4 decimals requires >5000 terms

    One of the many articles: http://mathforum.org/library/drmath/view/54456.html
  3. Standard member XanthosNZ
    Cancerous Bus Crash
    22 Aug '05 09:47
    I like e^(i*pi) + 1 = 0 myself. The most elegant mathmatical formula I've seen.
  4. 22 Aug '05 10:30
    Originally posted by XanthosNZ
    I like e^(i*pi) + 1 = 0 myself. The most elegant mathmatical formula I've seen.
    True, a beautiful combination of the most important numbers in C. But pi is already fascinating enough in R. It is not only 'irrational', but has a 'transcendental' aspect too. Unlike many other irrational numbers, like sqrt(2) for instance, it cannot be found as the result of a polynomial equation with integer coëfficients.
  5. Standard member XanthosNZ
    Cancerous Bus Crash
    22 Aug '05 12:32
    Originally posted by Mephisto2
    True, a beautiful combination of the most important numbers in C. But pi is already fascinating enough in R. It is not only 'irrational', but has a 'transcendental' aspect too. Unlike many other irrational numbers, like sqrt(2) for instance, it cannot be found as the result of a polynomial equation with integer coëfficients.
    After reading this page:

    http://en.wikipedia.org/wiki/Transcendental_number

    My head hurts.
  6. Standard member Nemesio
    Ursulakantor
    22 Aug '05 20:19
    Originally posted by Mephisto2
    True, a beautiful combination of the most important numbers in C. But pi is already fascinating enough in R. It is not only 'irrational', but has a 'transcendental' aspect too. Unlike many other irrational numbers, like sqrt(2) for instance, it cannot be found as the result of a polynomial equation with integer coëfficients.
    I, too, do not understand the concept of 'transcendental.' Can you
    walk me through it?

    I undersand ir/rational.

    Nemesio
  7. Standard member Nemesio
    Ursulakantor
    22 Aug '05 20:26
    Originally posted by Mephisto2
    is coverging extremely slowly, hence not very practical. An accuracy of 4 decimals requires >5000 terms
    This is amazing. It converges *that* slowly? The 5000th term
    would be something like ... - 1/9995 + 1/9997 - 1/9999 + 1/10001.

    That subtracting of 1/9999 (.0001000100010001...) gives the iteration
    stability only to the fourth digit? Incredible.

    Pi is an amazing number, no doubt.

    Nemesio
  8. Standard member PBE6
    Bananarama
    22 Aug '05 20:28
    Originally posted by Nemesio
    I, too, do not understand the concept of 'transcendental.' Can you
    walk me through it?

    I undersand ir/rational.

    Nemesio
    One way to think about transcendental numbers is graphically. Transcendental numbers cannot be generated by finding solutions to polynomials with integer coefficients, so graphs of f(x)=0 will never intersect the x-axis at a transcendental number. Never. Never ever. You can't bend them, twist them, stretch them, move them up or down, or rework them in any way using integer coefficients that will make that happen. You'll miss.

    I'm not sure of the origin of the use of the word "transcendental" to describe these numbers, but they kind of "transcend" attempts at solution using common sense numbers (integers).
  9. 22 Aug '05 20:51
    Originally posted by Nemesio
    I, too, do not understand the concept of 'transcendental.' Can you
    walk me through it?

    I undersand ir/rational.

    Nemesio
    I recommend to consult literature if you want to know more (which would very soon exceed my knowledge on the subject). I can just add that a transcendental function cannot be written using only a finite number of elementary operations (addition, multiplication, additive or multiplicative inverses, integer root s). It is neither a polynomial function nor an algebraic function. Examples are all trigonometric functions and logartmic functions.
    In analogy, a transcendental number is a real (and irrational) number that cannnot be obtained as a zero of a polynomial with rational coefficients. Their number is uncountable (as opposed to rational and non-transcendental irrational numbers).

    Coming back to pi, it is because pi is a transcendental (and not algebraic) number that it is impossible to 'square a circle', using only a compass and a ruler. This is what made me use the word transcendental in the first place.
  10. Subscriber sonhouse
    Fast and Curious
    22 Aug '05 21:20
    Originally posted by Nemesio
    I recently learned that:

    Pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...&c &c

    This seems unbelievably fascinating to me. First, because pi which is utterally
    irrational would have such a seemingly elegant mathematical defintion, and
    second that this was discovered at all.

    Does any mathematician want to talk about this defintion, how it came about,
    how it is applied?

    Nemesio
    Well then pi must equal 4*(1- 1/3 + 1/5 -1/7 + 1/9 -1/11......)
  11. 22 Aug '05 21:57
    Originally posted by sonhouse
    Well then pi must equal 4*(1- 1/3 + 1/5 -1/7 + 1/9 -1/11......)
    now, THAT is transcendental
  12. Standard member Bowmann
    Non-Subscriber
    22 Aug '05 22:04
    Originally posted by sonhouse
    Well then pi must equal 4*(1- 1/3 + 1/5 -1/7 + 1/9 -1/11......)
    Which is roughly 3. Amazing.
  13. Standard member Palynka
    Upward Spiral
    23 Aug '05 02:59 / 1 edit
    Originally posted by Bowmann
    Which is roughly 3.
    Or not.
  14. 23 Aug '05 06:59
    Originally posted by Nemesio
    That subtracting of 1/9999 (.0001000100010001...) gives the iteration
    stability only to the fourth digit? Incredible.
    Yes because the terms before and after this are approximately equal in magnituted and have opposite signs.
  15. 23 Aug '05 10:12
    Originally posted by Palynka
    Or not.
    Or is.