Expert: Ahmed Salami - 6/16/2004

Question
Hello,
I'd be very thankful if you answer the following two questions:
1-Is there any formula to enable you to calculate the result of a factorial without multiplying by the numbers one by one?(for example what's the result of 7654789!)
2-Is there any formula to enable you to calculate the perimeter of an ellipse having its two diameters?
Thanks a lot for your soon reply  

Answer
Hi teacher,
(1)Yes, there is Stirling's formula which allows us to do that. It states, for large n
n! ~ sqrt(2#n).n^n.e^(-n)
where # = pi
As for the example, we have
7654789! ~ sqrt(2#.7654789). 7654789^7654789. e^-7654789 = S
The trick is to use logs
log S = 1/2log(15309578) + 1/2log# + 7654789log(7654789) -7654789log(e)
     = 3.59248161 + 0.248574936 + 52695056.32 - 3324432.623
     = 49370627.54
from which
S = 3.467 x 10^49370627, a number with 49370627 digits.

(2) I cannot say that there is a direct formula to do this one. This can, however, be done with some techniques of calculus depending on the values of the diameters.
The equation of an ellipse is (x/a)^2 + (y/b)^2 = 1
where a and b are the semi-major and semi-minor axes respectively.
You would have to build up a definite integral to be evaluated. The perimeter can now be gotten from tables published under the topic of elliptic functions which you would do well by reviewing.

I hope i have helped you.
Regards.