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.
But ... we have calculators in our pockets. I'm not sure how you get to do so many fermi estimates that it's actually time-efficient to do the arythmetic in your head instead of the much more accurate and faster calculator on your phone. Doing it in your head isn't faster and just adds additional uncertainties on top of the uncertainties in each of your inputs.
Yeah, this is a fair point!
Let's see -- a median Fermi estimate might involve multiplying 5 things together. If it takes 7 seconds to pull up my calculator app, and that lets me do a perfectly accurate operation every second instead of slightly-error-prone operation every two seconds, then using the calculator gives me a 100% accurate answer in 12sec instead of a five-times-slightly-inaccurate answer in 10sec.
I still feel skeptical for some reason, but that's probably just status quo bias. This seems like a reasonable tradeoff. I'll try it for a month and see how it goes!
Personally I am very slow at typing on my phone. Always have been, I'm old at heart. I also find inputting values with large exponents to be inconvenient and slow. So if I'm not by my laptop, I tend to do quick calculations in my head, then only use my phone to double check order of magnitude. I actually prefer pencil and paper to typing on my phone a lot of the time.
Edit to add: My wife prefers to use her calculator, and is usually a tad slower than me, but does catch errors I miss, maybe one time in 10?
Huh. I usually do it in my head. Maybe I should try doing it with a calculator. I think it might actually be faster in my head.
Excuse you, you mean 6^(+/- 35) !
Oof, tracking the log6 instead of the log10 is such a horrifying idea I didn't even think of it. I guess you could do that, though! I guess. Ew. I love it.