Now that most people are carrying around a powerful computer in their pocket, what use is it to be able to do math in your head?

Here’s something I’ve noticed lately: **being able to do quick approximations in mid-conversation is a superpower**.

When I’m on Zoom with a client, I can’t say “Excuse me a second. Something you said gave me an idea, and I’d like to pull out my calculator app.” Instead, I can say things like “That would require collecting four times as much data. Are you OK with that?”

There’s no advantage to being able to do calculations to six decimal places on the spot like Mr. Spock, and I can’t do that anyway. But being able to come up with one significant figure or even an order-of-magnitude approximation quickly keeps the conversation flowing.

I have never had a client say something like “Could you be more precise? You said between 10 and 15, and our project is only worth doing if the answer is more than 13.2.” If they *did* say something like that, I’d say “I will look at this more carefully offline and get back to you with a more precise answer.”

I’m combining two closely-related but separate skills here. One is the ability to simple calculations. The other is the ability to know **what** to calculate, how to do so-called Fermi problems. These problems are named after Enrico Fermi, someone who was known for being able to make rough estimates with little or no data.

A famous example of a Fermi problem is “How many piano tuners are there in New York?” I don’t know whether this goes back to Fermi himself, but it’s the kind of question he would ask. Of course nobody knows exactly how many piano tuners there are in New York, but you could guess about how many piano *owners* there are, how often a piano needs to be tuned, and how many tuners it would take to service this demand.

The piano tuner example is more complicated than the kinds of calculations I have to do on Zoom calls, but it may be the most well-known Fermi problem.

In my work with data privacy, for example, I often have to estimate how common some combination of personal characteristics is. Of course nobody should bet their company on guesses I pull out of the air, but it does help keep a conversation going if I can say on the spot whether something sounds like a privacy risk or not. If a project sounds feasible, then I go back and make things more precise.

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The thing I’ve noticed in people is that those who are not proficient in mental calculations just avoid the process.

They don’t whip out a calculator to see if the product they want to buy is a good deal, or if the investment will pay out above its cost, or if the probability of something is reasonable. Instead of mathematical reasoning, they rely on other heuristics — like social cues — for making decisions.

Math becomes a blind spot, not just something slightly slower.

I experienced something similar as my eyesight was deteriorating. I stopped relying on reading signs for navigating, slowing down so I can react to visual feedback that requires closer proximity, and I avoided physical activities that require precise or rapidly processing visual cues.

That’s a good point. If something takes too long, people will often simply not do it rather than do it slowly.

Being able to quickly come up with (the first significant figure and the order of magnitude) also helps you know when to raise the “I think we made a mistake in that calculation” flag, and decide that a double check is worth the effort.

Uses and Abuses: what about hasty mental calculations made by managers with little notion of the consequences

@Jonathan: Absolutely. Professors aren’t much better than students at preventing errors, but they’re much better at detecting errors.

When a professor makes a mistake, no big deal. When they say “Hmm, that can’t be right” pay attention. Then is the time to ask a question. “How do you know that can’t be right? This sounds important.”

@Alvaro: Snappy estimates are good but snap decisions are bad. If a calculation is important, then do it carefully. Even better, do it two or three different ways if you can.

I’ve found quick approximations to be invaluable in my engineering career, for double-checking the gross accuracy of a precise calculation as well as initial single-digit-accuracy calculations. Surely one contribution to this was having learned to use a slide rule, where one needs to calculate separately (either on paper, or ideally, mentally) the magnitude of the result.

I recently read “The Joy of X” by Steven Strogatz in which he discusses filling a bathtub. “If the cold water faucet fills it in a half hour and the hot water faucet fills it in an hour, how long will it take to fill the tub when they’re both running together?” He gives several ways to solve the problem, but also how to quickly find common-sense limits on the solution based on easy assumptions. With only the cold water faucet running it takes 30 minutes to fill. If the hot water faucet ran at the same rate as the cold, it would take half the time, 15 minutes, but since the hot runs at a lower rate, it will take longer than 15 minutes, but less than the 30 minutes with the cold water faucet only. Knowing this of course doesn’t guarantee that the exact answer is correct, but it does show if it’s within logical limits that are easily calculated mentally.

I have rarely if ever seen such approximations or ways to double-check an answer formally taught or discussed (except for dimensional analysis, and even that was perhaps not emphasized enough). Looking back, I feel lucky that any instructors or professors mentioned these in passing.

Yet another post to be tagged with #GristForMyBook on 21st Century Knowledge Management (e.g. your post on technical ‘boundary layers’, and many, many others.)

I’ll help crowdfund such a book, and buy ten (10) copies of the first run, (with sole proviso it’s not simply a compilation of existing posts.)