Posers and Puzzles
22 Aug 05
22 Aug 05
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
22 Aug 05
Originally posted by NemesioI am not a mathematician, but I have read a few things about it.
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
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
22 Aug 05
Originally posted by XanthosNZTrue, 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 like e^(i*pi) + 1 = 0 myself. The most elegant mathmatical formula I've seen.
22 Aug 05
Originally posted by Mephisto2After reading this page:
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.
http://en.wikipedia.org/wiki/Transcendental_number
My head hurts.
22 Aug 05
Originally posted by Mephisto2I, too, do not understand the concept of 'transcendental.' Can you
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.
walk me through it?
I undersand ir/rational.
Nemesio
22 Aug 05
Originally posted by Mephisto2This is amazing. It converges *that* slowly? The 5000th term
is coverging extremely slowly, hence not very practical. An accuracy of 4 decimals requires >5000 terms
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
22 Aug 05
Originally posted by NemesioOne 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, too, do not understand the concept of 'transcendental.' Can you
walk me through it?
I undersand ir/rational.
Nemesio
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).
22 Aug 05
Originally posted by NemesioI 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.
I, too, do not understand the concept of 'transcendental.' Can you
walk me through it?
I undersand ir/rational.
Nemesio
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.
22 Aug 05
Originally posted by NemesioWell then pi must equal 4*(1- 1/3 + 1/5 -1/7 + 1/9 -1/11......)
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