The moment when you add pi

livemint.com:

pi and below that, the total, $30. Above is one line: “He has been waiting for this moment his whole life." The moment, that is, when he got a chance to add pi to a bill and produce a nice round figure. Not given to a lot of us.

Then again, was it really pi? Then again, was it really a nice round figure—or, shall we say, nicer and rounder than other figures? Those are actually questions I find fascinating. To begin with, it wasn’t really pi he added as a tip. After all, pi is an “irrational" number.

That means it cannot be expressed as a fraction. Another way of saying that is that its decimal expansion never ends. In this bill, what the man added as a tip was $3.14—and 3.14 is, of course, the fraction 314/100.

So it is, as you no doubt know, close to but not actually pi. Accurate to the first 10 places after the decimal point, pi is 3.1415926536. Accurate to 18 places, it would be 3.141592653589793238.

And, of course, I could go on forever. So, 3.14 is indeed an approximation—a reasonably good one, but an approximation all the same. In passing, you would not call any of those approximations to pi a “nice round figure", I’m sure.

Why not, though? I’ll return to that. Before I do, a reminder of the great Srinivasa Ramanujan. If you know anything about him, you know that through his short life, he churned out endless exotic formulae that mathematicians are trying to understand even today, a century after he died.

Here’s a description of one of them that you can tap out on your nearest calculator: Add the square roots of 72, 90 and 80. Add 10 to that total. Of the result, take the natural logarithm (“ln"—never mind what it means, but your calculator will have a button, so press it).