I’m definitely not happy with my definition of “100% shorter” from last time. After all, if “100% longer” means “twice as long”, shouldn’t “100% shorter” mean “twice as short,” which is “half as long”?

To figure this out, let’s use “inverse-inches.” So if our rope is 60 inches long, it is 1/60 inverse-inches short. “100% shorter” is thus 1/60 + 1/60 = 1/30 inverse-inches short, or 30 inches long:

• 50% shorter – 1/60 + 1/120 = 3/120 inverse-inches – 40 inches
• 25% shorter – 1/60 + 1/240 = 5/240 inverse-inches – 48 inches
• 75% shorter – 1/60 + 3/240 = 7/240 inverse-inches – 34.3 inches
• 200% shorter – 1/60 + 2/60 = 3/60 inverse-inches – 20 inches
• 300% shorter – 1/60 + 3/60 = 4/60 inverse-inches – 15 inches

Now “200% shorter” is the same as “three times as short” (as described last time), which works the same way as longer/long, so I think we’ve got it. Still, it’s not exactly obvious.

In any case, length is obviously not the only measurement that this applies to; some of these seem more sensible than others:

• half as young is twice as old; twice as young is half as old
• half as many is twice as few; twice as few is half as many
• half as wide is twice as narrow; twice as narrow is half as wide
• half as big is twice as small; twice as small is half as big

(Of course, “big” has its own problems – what’s “twice as big” as a 13-inch television? Shall we double the lengths of the sides, the length of the 13-inch diagonal, the area of the screen, or the volume of the box? We’d need to be more precise.)

This stuff actually makes more sense when we use ratio measurements, as with our original measurement of speed. We measure how fast something is by measuring how many tasks are accomplished in an amount of time, e.g. miles per hour – bigger numbers are faster. To measure how slow something is, we measure the inverse, which is how much time it takes to do a number of tasks, e.g. seconds per lap – bigger numbers are slower.

So, if my task took 20% less time, then a previously 100-second task would now take 80 seconds. To measure how fast the task is, we invert the measurement – I improved the speed from 1/100 tasks per second to 1/80 tasks per second. In decimal, that’s 0.01 tasks per second to 0.0125 tasks per second. That’s an improvement of 0.0025 tasks per second, which is 25% of 0.01, so we’d say that the task runs 25% faster.

Ouch. My brain hurts. Since I’d have to justify why shaving 20 seconds off of 100 is “25% faster”, I should probably stick with “20% less time.”

In summary then, we should probably avoid using phrases that would require us to invert our preferred form of measurement. Even “150% longer” can be confusing (it means 2.5 times the length), so you might avoid phrases like that as well. Stick with language that is clear and unambiguous.