When doing Fermi estimates in my head, I sometimes lose track of powers of ten: ...that's three billion widgets, divided by fifty widgets per person is somewhat under a hundred million people, times $20 per person is somewhat under $2 billion... or, wait, did I drop a zero, should that have been somewhat under a billion people? Did I have three billion or thirty billion widgets? Shoot.
I find that it helps a lot to count on my fingers like a common third-grader! It's a lot harder to misplace fingers than to misplace the idea of a decimal point (probably because fingers... exist). Specifically, my calculations tend to go much more smoothly if I represent the of the accumulator on my fingers:
(I use "palm up/down" to represent the sign of the log, since it often flip-flops during the calculation.)
A couple of worked examples:
Is it worth joining this social movement? How many hours does it take to save a QALY?
Well, if there are 1M people... (1e6 people) and each person spends 10hr/wk... (1e7 hr/wk) for 2 years = 100 wk... (1e9 hr) and thereby we make 100M people's lives... (1e1 hr/life) 1% better... (1e3 hr/QAL) and a life is 100 years... (1e1 hr/QALY)
then that averages to 10hr/QALY.
How many potential rationalists are there in Seattle?
Well, if 1/1000 people are receptive to the Message... (1e-3/person) and there are 2M people in King County... (1e3.3 people) and half of those are in Seattle... (1e3 people)
then there are 1000 potential rationalists.
This will break down once in a while, when intermediate values wander outside the range of (or if you're weird), but it's served me quite well for a couple years.