Here are three rules of thumb for back-of-the-envelope estimates:

- Duff’s rule: Pi seconds is a nanocentury.
- Hopper’s rule: Light travels one foot in a nanosecond.
- Rule of 72: An investment at n% interest will double in 72/n years.

How might you use these? How accurate are they?

**Duff’s rule** comes in handy when converting from times measured in seconds to times measured on calendars. This may not sound useful, but it often happens in software. For example, if a task takes a second to complete, how long would it take to do it a billion times? Well, a billion seconds, obviously. But how long is that in familiar terms? Duff’s rule says a century is about 3.14 billion seconds, so a billion seconds would be something like 30 years.

How accurate is Duff’s rule? A year is 31,536,000 seconds, whereas Duff’s rule would estimate 31,415,927 seconds, so it underestimates the number of seconds in a year by about 0.4%.

**Hopper’s rule** is useful in electrical engineering. For example, you might need to know how long it would take a radio signal to travel between a transmitter and receiver. Hopper’s rule can explain why computer chip clock rates are not increasing. Electrical signals travel at some fraction of the speed of light, and current chip designs are limited by whether a signal can move across the chip during a clock cycle.

How accurate is Hopper’s rule? Light travels 299,792,458 meters per second. That corresponds to 0.983 feet per nanosecond, so Hopper’s rule overestimates by about 1.7%.

Here is a terrific video of Grace Hopper explaining Hopper’s rule to David Letterman, around 4:25. (Thanks Bill!)

The **Rule of 72** is obviously useful in financial estimates. For example, $1000 invested at 6% interest will become $2000 in 72/6 = 12 years.

How accurate is the rule of 72? The value of an initial investment *P* at time *t* with under a continuous interest rate *r* is *P* exp(*rt*). Solving exp(*rt*) = 2 for *t* gives *t* = log 2 / *r*. If we express *r* as a percentage, we have to multiply *t* by 100. This says that for continuously compounded interest, the rule of 72 would be exact if “72” were replaced with 100 log 2 = 69.3. So for continuous interest, the rule overestimates the doubling time by 0.72/log 2 or about 4%. So why use 72 rather than 69.3? There are two reasons. First, 72 is easy to work with mentally since it is divisible by lots of small integers. Second, interest is often compounded periodically — say annually or monthly — rather than continuously.

The doubling time is longer for investments with periodic interest rather than continuous interest. The overestimate from using 72 rather than 69.3 is partially canceled out by the accounting for the longer doubling time for periodic compounding and so 72 may work better than 69.3. Exactly how accurate the rule of 72 is for periodically compounded interest depends on the interest rate and the compounding period.

**Related post**: Bancroft’s rule (rule of thumb for estimating linear regression)

Grace Hopper gave a fun interview back in the early 1980’s on David Letterman. She explains Hopper’s rule, and even takes it a step further and shows Dave what a picosecond looks like.

Thanks, Bill. I updated the post to include the interview you pointed out.

Re: the video (at about 7:55): A picosecond is a trillionth, not quadrillionth of a second.

Thanks Bill for sharing that video, and John for adding the link. Made my day.

I know it’s “late”, but I suspect that Grace’s “nano-second” was actually 30cm long (0.3m) , rather than 12inches. There is a slight difference ;’) in which case, her nano-second is (by my reckoning) about 1/3 of 1% ‘off’.

My reasoning is that she was a Scientist, and she would have been familiar with the MKS (meters / kilograms / seconds) system. Indeed, the wikipedia article seems to back up this inference that she used 30cm (11.8in) rather than 1foot.

great site,

.h

the video is not available anymore, its account has been removed due to copyright issues 🙁

I just saw this page in a tweet from @CompSciFact

The video has been taken down, but a quick search brings up another copy:

http://www.youtube.com/watch?v=RZ0g5_NgRao

It’s funny that both the speed of sound and speed of light can be so readily approximated by “1 foot per x” – 1 foot per millisecond is a very common rule of thumb in acoustics, sound recording (microphone arrays) and live sound (speaker arrays).

I am rather dubious that those the takedown notices are valid.

But they smell like they are specifically targeting this blog. I wonder what’s triggering them?

the video has been removed by user..

One for fretted musical instrument makers (for the modern equal-tempered 12-notes-per-octave scale) is the rule of 18. This page gives a more-than-comprehensive explanation:

http://www.liutaiomottola.com/formulae/fret.htm

For the David Letterman video, in the era of DCMA takedown notices, URL’s become outdated quickly but the videos keep getting put back up. Searching Youtube for David Letterman Grace Hopper should usually work.