Somehow, I suspect I wouldn’t survive long on the frontier.
Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.
And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful Penguin Book of Curious and Interesting Numbers, by David Wells:
I decided to play around with this product a bit. After all, what are products for, if not playing around?
(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)
I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:
Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.
The left side of that equation is π/2. It’s roughly 1.57.
The right side of that equation, however, is a product of many numbers—all of them below 1.
What happens when you multiply two numbers smaller than 1? You get another number smaller than 1.
How the heck could that equal 1.57?