- 22 Aug '05 07:24I 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 07:53

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

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 10:30

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.*Originally posted by XanthosNZ***I like e^(i*pi) + 1 = 0 myself. The most elegant mathmatical formula I've seen.** - 22 Aug '05 12:32

After reading this page:*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.**

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

My head hurts. - 22 Aug '05 20:19

I, too, do not understand the concept of 'transcendental.' Can you*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.**

walk me through it?

I undersand ir/rational.

Nemesio - 22 Aug '05 20:26

This is amazing. It converges *that* slowly? The 5000th term*Originally posted by Mephisto2***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 20:28

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.*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'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 20:51

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.*Originally posted by Nemesio***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 21:20

Well then pi must equal 4*(1- 1/3 + 1/5 -1/7 + 1/9 -1/11......)*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