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

1. Duff’s rule: Pi seconds is a nanocentury.
2. Hopper’s rule: Light travels one foot in a nanosecond.
3. 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%.

There’s a video of Grace Hopper explaining Hopper’s rule to David Letterman, but it keeps getting taken down.

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)