**Duff’s rule** says that a nanocentury is about π seconds. Assuming a year is 365.25 days, there are 3,155,760,000 seconds in a century. So a nanocentury, one billionth of a century, is 3.15576 seconds, roughly π seconds.

This odd fact is surprisingly useful in back-of-the-envelope calculations. In order to determine whether something is computationally feasible, you have to go from how much work can be done in a second to how much work could be done on calendar time scales. For example, suppose some operation is going to take 10^{15} steps and you can carry out 10^{6} operations per second. How long would that take? Obviously 10^{9} seconds, but how long is that in familiar units of time? According to Duff’s rule it’s about a third of a century so about 30 years.

Since the square root of 10 is approximately π, you could say there are about square root of 10 seconds in a nanocentury. In fact, the square root of 10 is 3.162 and so it’s closer to the number of seconds in a nanocentury than π is. The advantage to square root of 10 is that it is exactly half an order of magnitude, 10^{1/2}. So you could say a century is 9.5 orders of magnitude longer than a second, or a year is 7.5 orders of magnitude longer than a second. Or perhaps more memorably,

A year is about 30 megaseconds.

If you interpret “mega” as 2^{20} rather than 10^{6} this approximation gets even better. (Technically, this would be 30 mebiseconds. SI distinguishes “mega” = 10^{6} = 1,000,000 from “mebi” = 2^{20} = 1,048,576, though the latter isn’t widely used. Most people have either never heard of the new prefixes like “mebi” or think they sound silly and prefer the ambiguity of using “mega” to mean two slightly different things. See Kibi, mebi, gibi.)

I wrote briefly about Duff’s rule a while back in the post Three rules of thumb. That post also includes a great video of Grace Hopper explaining to David Letterman her rule of thumb that light travels about one foot in a nanosecond.