Here’s a fun formula for Pi involving an infinite product, known as Wallis’ Formula:
(Pi/2) = (2*2)(4*4)(6*6)/(1*3)(3*5)(5*7)
It is somewhat surprising that when you pull out every other pair of terms, you get a completely different kind of number!
Sqrt[2] = (2*2)(6*6)(10*10)/(1*3)(5*7)(9*11)
Presentation Suggestions:
You can also start with the infinite product, and ask if student can guess what it converges to, before you tell them the answer.
The Math Behind the Fact:
There is an infinite product formula for the sine function which yields Wallis’ formula as a consequence. Infinite products are defined as the limit of the partial products, which are finite. This is similar to the way we define infinite sums!
How to Cite this Page:
Su, Francis E., et al. “Wallis’ Formula.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
S. Vandervelde, Mathematics and Informatics Quarterly, 9 (1999), pp. 64-69.
Fun Fact suggested by:
Sam Vandervelde