I wonder if this is related, but imperial measurements with a 5 in the numerator (and a power of two in the denominator) are generally just under a power of two number of millimeters.

The reason is fun, and as far as I know, historically unintentional. To convert from 5/(2^n) inches to mm, we multiply by 25.4 mm/in. So we get 5*25.4/(2^n) mm, or 127/(2^n) mm. This is just under (2^7)/(2^n) mm, which simplifies to 2^(7 - n) mm.

This is actually super handy if you're a maker in North America, and you want to use metric in CAD, but source local hardware. Stock up on 5/16" and 5/8" bolts, and just slap 8 mm and 16 mm holes in your designs, and your bolts will fit with just a little bit of slop.

So the error is 1.6%. Acceptable for everyday hardware I guess.

My favorite is 1 mile = phi kilometers with <1% error

That one’s useful too. If you know a few Fibonacci numbers you can convert miles to kilometres and vice versa with ease.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …

21 km is ~13 miles, 13 km is ~8 miles, etc.

A 26 mile marathon? Must be ~42km.

Same for speed limits too; 34 mph is ~55 kmh

I use that approximation, via the Fibonacci sequence, to translate between miles and km. 13 miles ~ 21 km (actually 20.921470).

My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.

... you all realize that phi is barely a better approximation than 8/5, right? 1.6 vs 1.609 (km in a mile) vs 1.618?

(8/5)/(1 mile/1 km) = 0.9942; (1 mile/1 km)/phi = 0.9946. You're making things way harder on yourself for essentially no improvement in precision, especially when you're just rounding to the nearest whole number.

That tells me you haven't memorized the first ten or twelve terms of the Fibonacci sequence.

1 km = 5 furlong, with < 1% error.