Let's tour some fun ways to do mental math to various precisions.

I like doing order of magnitude problems, so own a wide array of books on the subject, including books full of fun estimation problems. One of the books I own is Maths On The Back Of An Envelope by Rob Eastaway. In it, he suggests a form of estimation he brands zequals, i.e. you round numbers to only a single significant digit and do your calculation that way.¹² This form of "ruthless" approximation is meant to make it easy to do mental arithmetic.

The branding is cute. I think trying to round things to a single significant digit is something people tend to do pretty naturally when they are estimating. However, thinking about it again, I think there is a neat way to get the same or better precision with even less of the fuss. We should use quarter orders of magnitude, or work directly in decibels. Let's build up to that.

Order of Magnitude

When you get into Fermi problems you often start by simply tracking the orders of magnitude, i.e. you round every number to its nearest power of 10 and only keep track of those. This makes for very speedy estimates, but the resolution is obviously only good to about an order of magnitude. This works great for trying to answer questions like my son asked the other day: "If we suddenly got an extra electron on each of our atoms, what would happen?"³

We don't really care about accuracy that is better than a factor of 10, we are more interested if it would be like a punch to the gut, a bomb going off, or Armageddon. Order of magnitude math is good for this kind of thing, and easy to do mentally or on paper. You only need to track the powers of 10. Multiplication and division become as easy as addition and subtraction:

One Significant Digit

The natural next step up in precision brings us back to Eastman's zequals, or one-significant-digit arithmetic. Here we'll reduce every number to just its leading significant digit.

For example, the speed of light: $299792458\text{ m/s}$ becomes simply $3\times {10}^8\text{ m/s}$.

The full multiplication table for this system is the one we all learned in grade school:

	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$1$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
$2$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$	$18$
$3$	$3$	$6$	$9$	$12$	$15$	$18$	$21$	$24$	$27$
$4$	$4$	$8$	$12$	$16$	$20$	$24$	$28$	$32$	$36$
$5$	$5$	$10$	$15$	$20$	$25$	$30$	$35$	$40$	$45$
$6$	$6$	$12$	$18$	$24$	$30$	$36$	$42$	$48$	$54$
$7$	$7$	$14$	$21$	$28$	$35$	$42$	$49$	$56$	$63$
$8$	$8$	$16$	$24$	$32$	$40$	$48$	$56$	$64$	$72$
$9$	$9$	$18$	$27$	$36$	$45$	$54$	$63$	$72$	$81$

While you have likely had this table memorized since you were about five years old, we can admit that it is somewhat complicated. It requires memorizing 100 entries. And while we can quickly do single digit multiplications, division is much harder. How many people know what 1/8 is, even to one significant digit? Granted, with the extra costs we've got an increased precision.

Unfortunately, because most of the type of math we do when we do order of magnitude problems is multiplication and division, the relative error in our system is set by the smallest multiplicative factor between symbols. In this case, the gap between 1 and 2 is quite large, and this system struggles to maintain accuracy to within a factor of 2 in general. Here we are using 10 symbols per decade but only get accuracy to a factor of two.

Two Significant Digits

If we wanted to have even greater precision, we could do our calculations to two significant digits. This starts to be beyond most people's capability for what they can do in their head. If not for multiplication, certainly for division. It requires use of 100 symbols, though again, we have a lot of practice with these symbols and how they multiply. Doing two significant digit arithmetic is a lot easier to do on paper and it what I would typically use back in my undergrad physics classes. Keeping around two significant digits ensures that our answers are good to a factor of 1.1 or so, i.e. 10%.

Can we do better?

More Exotic Logarithmic Systems

We can get higher accuracy with fewer symbols if we distribute our symbols evenly on a logarithmic scale.^{1 2}

One - Few - Ten

One of the biggest bang-for-your-buck type systems is one I like to call one-few-ten. If we simply track half-order-of-magnitude we can, with only two symbols, achieve ~30% relative errors.

Now, instead of rounding each number to the nearest power of 10, you round each number to the nearest half-power of 10. This sounds like it might be complicated but ends up being very simple in practice.

When you play with slide rules, you quickly really internalize how close $\pi$ is to the square root of 10.

So, in practice you simply round each number to either the nearest power of 10 or $\pi$ times the nearest power of 10.

For example:

Looking at the Logarithmic circular dial above, the precise rounding points are ${10}^{1/4}\approx 1.78$ and ${10}^{3/4}\approx 5.6$, which at the level of precision we are dealing with you could call 2 and 6. To use the One-Few-Ten system then, if the first digit of a number is between 2 and 6, you call it $\pi$ times the relevant power of 10, and otherwise you just round it to the nearest power of 10. Two $\pi$s make another ten: ${\pi }^2\approx 10$. The 'arithmetic' is quite simple:

	$1$	$\pi$
$1$	$1$	$\pi$
$\pi$	$\pi$	$10$

So, this doesn't really add any sort of mental burden, but increases our accuracy from being good to only an order of magnitude or factor of 10, to being good to a factor of $\pi$.

I think everyone should be done one-few-ten type arithmetic as a default, and wish it was more popular. Can we do better than this without increasing the mental burden too much?

Quarter-Orders-of-Magnitude

Why yes, I believe we can. Let's use quarter orders-of-magnitude! If we split the decade into four pieces, we can achieve a relative accuracy of ${10}^{1/4}\sim 1.8$, better than the factor of 2 we got from the zequals system, and only using 4 symbols instead of 10.

To make this even easier to intuit, I suggest using the symbols, $1,\tau ,\pi ,$ and $\tau \pi$ (pronounced "tripi") Here $\pi$ is nearly equal to the $\pi$ we know and love, and $\tau ={10}^{1/4}\approx 1.78$ while $\tau \pi ={10}^{3/4}\approx 5.62$. In terms of these symbols, the arithmetic table makes a lot of intuitive sense, you simply add the legs, with 4 legs equaling a factor of 10.

	$1$	$\tau$	$\pi$	$\tau \pi$
$1$	$1$	$\tau$	$\pi$	$\tau \pi$
$\tau$	$\tau$	$\pi$	$\tau \pi$	$10$
$\pi$	$\pi$	$\tau \pi$	$10$	$\tau 10$
$\tau \pi$	$\tau \pi$	$10$	$\tau 10$	$\pi 10$

As can be seen from the dial, the appropriate thresholds to round at are ${10}^{1/8}\approx 1.33,{10}^{3/8}\approx 2.37,{10}^{5/8}\approx 4.22,$ and ${10}^{7/8}\approx 7.50$. As a bit of a mnemonic to remember these thresholds, remember that we are doing things in quarters, the first threshold is 1.3, and 1+3 = 4, then 2.4 and 4.2 which are opposites of one another and finally 7.5 which looks a lot like 3/4. Now you can have even better precision than you get from single-significant-digit arithmetic, but without all of the mental cost. Division is also very doable, again, you need only count the legs. $10/\tau =\tau \pi ,\tau \pi /\pi =\tau$, etc.

I haven't seen this "quarters" system described elsewhere. I've started using it myself and I think it has a lot of promise. It seems to be a very good compromise between speed, ease of use, and accuracy.

Decibels

As a final, only slightly outlandish proposal, I'll suggest that if we wanted a system that was accurate to 25%, we could simply use decibels. This amounts to expressing each number as 10 times a power of 10:

If we round this to the nearest integer, we end up with an approximation that is good to ${10}^{1/10}\approx 25\mathrm{\%}$.

We are used to doing calculations with numbers in scientific notation. This already separates a number into two pieces, its power of 10 and its significand, or part that is left over. Doing arithmetic to two significant digits means writing every number in scientific notation with two significant digits. If we convert this into decibels, that power of 10 now becomes the 10s place for the number, while the units place represents what fraction of a decade the significant represents. This ends up being roughly as much ink on the page as we would have used otherwise.³

Granted, this does require memorizing how to convert numbers to decibels, but this isn't all that difficult, this is a one time cost. As pointed out by John Cook there is a clever way to approximately find the integer decibel values if you forget them. First make a list of the first powers of two:

then lexigraphically order them: then insert decimals after the first digits: As you can see, this very well approximates the locations of the integer decibels on the scale above.

Fortunately, this turns multiplication and division into simple addition and subtraction of integers, something we are much better primed to do. If we wanted to match the precision of two-significant-digit arithmetic we would need to track the nearest half decibel as well, but even this is pretty easy to do in our head. Quick, what is 4.5 + 8? Now try to do 2.8 * 6.3 to two sigfigs. How about 4.5 - 8 and 2.8 / 6.3? Which of those was easier? I think doing arithmetic with half integers is a lot easier, especially subtraction compared to division.

Conclusion

Try out the quarters thing. Its fun and gives as good precision as single-significant-digit arithmetic. If you are feeling more adventurous and want higher precision, try decibels.