roundness of numbers is complicated

87.5 is totally a round number

Inkhaven participant and YouTuber¹ Signore Galilei has put forth a take on what it means for a number to be round, which has inspired me to expound upon my feelings regarding number roundness. Galilei’s take is not a bad first pass at all, I do not mean any shade towards it at all—it’s just that I’ve refined my intuition about number roundness too far and will probably never be satisfied by an attempt to quantify it.

Okay, maybe that’s not quite right. The p-adic valuation is pretty good:

\(\begin{align} &\nu_p\;(\,n\,) \;= \sup\{k \in \mathbb{N} : p^k \mathop{|} n\}\\ &\nu_p \left(\frac{a}{b}\right) = \nu_p(a) - \nu_p(b) \end{align}\)

In general sup means “least upper bound”, which is sort of like a maximum except it works on (for example) open sets—but in this case, what it means is “April didn’t feel like separately writing out that ν(0) = ∞.” The “|” means “divides” or “is a factor of”.

Basically, this asks “how many zeroes does n end with if expressed in base p”, with some nuances. For one, we can say 0.1 ends in -1 zeroes. I think this makes sense: 1 ends with no zeroes, dividing by 10 takes away a zero, so 0.1 ends with -1 zeroes. For two, 0 itself ends with infinite zeroes.

I have achieved the level of enlightenment where it makes perfect sense to me to say that 0 ends with specifically positive infinite zeroes—look at that supremum definition up there, doesn't it feel natural? Can’t you divide a ten from 0 as many times as you like, and still get an integer? But I will concede that viewing 0 as having negative infinity zeroes also makes sense from certain points of view².

For three, 1/3 ends with no zeroes, just as one does, though it is not divided by 10⁰. Consider: you can multiply by three a bunch of times, and this adds no zeroes. You can then divide by three a bunch of times, and this removes no zeroes. So we will say that dividing by three, in general, does not remove zeroes (in base ten). A little weird, negative infinity seems like a possibly more intuitive choice for 0.333…, but oh well.

You may then ask: doesn’t 10/2 = 5 remove a zero? This is why we use the variable p. It gets uglier when you aren’t particularly interested in one particular prime base³—a factor of ten is made by pairing up a factor of two and a factor of five, and you’d sort of need to bookkeep those separately.

The p-adic valuation works well enough for me to find it satisfying. It’s a mathematically elegant notion that ends up coming up naturally in some fascinating theorems, and I can be convinced that it’s sort of a notion of roundness. Sure, n² is twice as round as n. Sure, roundness is basically just how many factors of p I can fit into a number. Sure, zero is kind of like p^∞⁴, and thus rounder than any other number.

It’s really impressively round. Most people would say only eight comes anywhere close. But I’m nοt mοst peοple—I wοuld argue seventy actually manages tο beat zerο.

But as nice as the p-adic valuation is, and as much as it is pretty closely related to roundness, I don’t think it comes close to a full explanation of the pattern underlying my intuitions about roundness. It’s a good account of one particular aspect of roundness.

A full explanation will definitely not come in the form of a nice and simple function from the natural numbers to the reals,

\(\mathrm{roundness}: \mathbb{N} \to \mathbb{R}.\)

Intuitions about roundness

Let’s start with some intuitions that I expect to seem reasonably obvious to most people:

• Powers of ten are round

• (Small?) multiples of 5 or 50 or 500 are round, but not as much as the powers of ten

• 25 and 75 are kinda round, but maybe not as much as 50

• Multiples of 20 are sorta round, probably less-so than 50, not too straightforward to compare with 25 or 75

• Prime numbers are not very round, except perhaps two and five

• Even numbers are rounder than odd ones

Some intuitions that Signore Galilei has:

• “30 should be more round than 3 by the same amount that 70 is more round than 7”

• It is at least fairly reasonable to regard all powers of ten as perfectly round

• 26 and 28 are similarly round

• 64 is mostly only especially round for computer programmers

• 675 and 875 are similarly round

A lot of these make sense, and I grant that I have enough computer programmer in me that I might be unusually fond of powers of two. Nonetheless, it is time to start quibbling until I have conveyed as much about my own brain’s idiosyncratic concept of “number roundness” as I can.

I think I have an intuition that multiples of three can be a contributing factor. Like, consider 24. That number is hella round. It’s two dozen, the number of hours in a day, eight by three. You can tell me that 28 is no rounder than 26, and I’ll frown and say that I think there is a meaningful difference, even if it’s not a large one. The idea that 26 or even 28 are as round as 24, though? Ridiculous. It could be the confluence of the three and eight together, more than either on its own, that sets 24 apart, but there’s definitely something up with 24. And all the multiples of 24 feel noticeably round to me up to like, 168.

Similarly, 30 seems much more round than 70 does. This is for a few reasons, some of which we’ll get to later, but one point is that a factor of three does more favors than a factor of seven. The lack of attention paid to numbers like 24 or 30 or 72 is a key part of why I’m not totally satisfied with Signore Galilei’s approach.

But I think it’s reasonable anyways! Let’s get into it.

The Signore Solution

Signore Galilei starts with this trial definition:

For a number n, find the smallest power of 10 m that’s larger than n, and find the largest factor k of n that’s a factor of a power of 10. The “Base 10 roundness” of n is then log_m k.

He then takes issue with the emphasis this places on multiples of four or 125, using the specific test cases of 26 vs. 28 and 675 vs. 875. His theory is that, since four doesn’t go into ten evenly, but goes into 100 twenty-five times, there are too many distinct endings of a multiple of four you’d have to remember. Similarly, 125 goes into 1000 eight times, and you can only remember at most seven endings. So, he says, it is better to pick the largest factor k only among those such that k goes into the smallest power of ten which k is a factor of fewer than eight times. That is, k only counts if there are fewer than eight distinct ways a multiple of k can end in base ten, if you look at however many terminal digits are necessary to determine whether a number is a multiple of k or not.

For a number n, find the smallest power of 10 m that’s bigger than or equal to n, and find the largest factor k of n that’s a factor of a power of 10 l, such that l/k < 8. The “Base 10 roundness” of n is then log_m k.

It’s a pretty good try, I'm not sure I buy it. Like, do you know what percentage 5/7 is? It’s 71%. Or 71.(428571) repeating%, if you want to be exact, which you shouldn’t. But like, okay, suppose you only care to know the n/7s to the nearest percent. Do you really want to remember all seven of those? And that’s if you count 0%, you already know 0%, you only actually need to remember six new things. Maybe if sevenths came up more you would feel like remembering those?

On the other hand, I can, actually, basically recognize the eighths on sight, though I guess I could be weird for that. The thing is, you don’t actually have to learn eight endings. You have to learn four: 12.5%, 37.5%, 62.5%, and 87.5%. This is because you already have 0%, 25%, 50%, and 75% down pat.

No, the hard ones are the sixteenths. If I ask myself what 1/16 is, it takes a moment to come up with 0.0625—I think the initial zero makes it a little harder to track in my head, for some reason. I might have slightly less trouble coming up with 6.25%.

And the multiples of 0.0625? Well, the ones that are multiples of 1/8 are very recognizable. For some reason I find 0.3125 in particular sort of recognizable as a multiple of 1/16, though it would take me a second to tell you that it’s five sixteenths. The others I’m not so good at.

But the thing is, I don’t actually need to be able to perfectly recognize them for it to contribute to my sense of how round a number is. If you show me a proportion with four decimal places, or a percentage with two, and the last two digits are 25 or 75… I might recognize it, or if it’s something like 0.5375 I might recognize it as definitely not a multiple of 0.0625, or I might be not too sure in either direction, but it would definitely factor⁵ into how I relate to the number and how round it feels.

Now, the multiples of four. If you ask me “April, 76 or 78, which one’s the multiple of four?” I will, after not too long, be able to orient myself around either 72 or 80 and tell you that it’s 76. And I think this does make me feel like 76 is, if very slightly, more round than 78. I definitely feel like 92 is rounder than 94—though it’s possible that that one has to do with letter grade cutoffs occurring at (100 - 8n)% in some of my schools⁶? Something like 52 doesn't stand out to me as appreciably rounder than 54, though at 56 I can immediately notice it’s 60-4, which helps. (And also, 56 is a multiple of eight.)

That all being said. While I do kinda feel that Galilei unfairly maligns the eighths, or the multiples of 125 or of 12.5% or however you wanna talk about that set of numbers—America even use to have a coin, the bit, which was worth 12.5¢!—I do think, other than that, this is a reasonable take on a notion of “base ten roundness”.

Why do I think that, despite all my quibbles?

We don’t actually think in base ten

Which is more round, twelve or fifteen?

Well, they’re both multiples of three, but who cares, that’s not a factor of ten. Under the “greatest common factor with a power of the base you're working in” idea, what’s relevant is that twelve is a multiple of four—or a maybe we only even care that it’s a multiple of two—and fifteen is a multiple of five. And then, since log_ten(two) < log_ten(five), we say fifteen is rounder. An essentially identical analysis applies to 38 and 35, to 94 and 95, to 64 or 66 and 65.

I think I want to give a totally different response, when asked to weigh the roundness of twelve and fifteen. It’s not, unfortunately, one that invites summarization with a single simple cool math formula, or even with a simple family of formulae parameterized by what base you’re using and how large your working memory is.

In return, I want to ask: are we counting eggs, donuts, or inches? Or are we counting seconds, minutes, or degrees⁷?

Similarly, consider 25 and 30. Are we counting cents or percentage points? Or are we counting time? Even within money, whether 20 or 25 is rounder depends on whether we're talking cents or dollars; even within time, whether 12 or 15 is rounder depends on whether we’re counting minutes or hours.

April! This is cheating, you’re bringing number systems with mixed radices that make them not really base ten into this!

Thank you for asking. I think this is basically a correct response, though it’s maybe a little more complicated than that. This is why I think Signore Galilei has come up with a pretty decent notion of base ten roundness—like, sure, maybe I think 100 is a horribly⁸ unround number for counting minutes, but this is because minutes aren’t really meant for counting in base ten. And even if we do insist on counting them in base ten, we still care that 60 minutes is an hour and 120 is two.

Anyways, mixed radices. Consider the notation:

\(07{:}23{:}56.\)

This is a common way to represent a duration, or a time of day⁹. The 56 represents 56 seconds. The 23 represents, not 23 100-second durations, but 23 minutes, each sixty seconds. The 07 represents seven hours, each sixty minutes.

So, instead of being able to say something simple like “the n^th digit from the right, zero-indexed, represents 10ⁿ seconds,” we kind of need to go one by one. The zeroth digit represents seconds, the first represents 10 seconds, the next 6*10 = 60 seconds or a minute, then 10*(6*10) = 600 seconds or 60 minutes, 6*10*(6*10) = 3600 seconds, 60 minutes, or an hour. From there each new digit is just ten times the previous. The first (10 seconds) and third (10 minutes) “digits” only range from 0-5 instead of 0-9, because the subsequent places only have six times the value they do, not ten.

This digit-by-digit framing may not quite be the right way to think of this, though. It can get a little more complicated:

\(4\mathop{\mathrm{days,}}07{:}23{:}56.\)

Okay, suppose we’re not just counting up hours forever the way that Livesplit does, we split off 24 hours into a day. Consider the one-hour place. It goes from 0 to 9, and then you carry a 10-hour and you go from 0 to 9 again, and you carry again and go from 0 to… well, as soon as you hit 4 you need to carry off four hours and two ten-hours to get one day. So this is worse behaved in a new way.

Here’s what you should probably do: throw out digits entirely. There is a seconds place, a minutes place, and an hours place. These represent 60⁰ seconds, 60¹ seconds, and 60² seconds. Each place contains one sexagesit, which is like a unit, bit, nat, trit, or digit except there’s sixty instead of one, two, e, three, or ten. Confusingly¹⁰, for our sixty necessary sexagesit numerals, we’ve chosen the symbols “00”, “01”, “02”, …, “57”, “58”, and “59”. But don’t start to think those are composed of two digits! The first¹¹ one only even stores a sexit of information.

Arguably, that’s just base sixty and isn’t even mixed radix yet! It only becomes mixed radix once we add the days place, which corresponds to only 24 hours, not sixty.¹²

Anyways, this is why 30, 45, or 60 are sometimes far, far rounder than 25 or 50—though we can, at least, agree on 75.

Conclusion

Are we just screwed, then?

Well, maybe a little. Really, even when you think you’re using base ten, are you really? Which is rounder, a hundred million or one billion? I think one billion. It’s a nice and convenient power of a thousand, the true radix of our number system—but seriously, I could be convinced that, say, half a billion is rounder than a hundred million is.

In general, when you're working with metric, a thousand is substantially rounder than a hundred. Unless you’re counting centimeters, and then it’s the other way around.

And one hundred thousand and ten million are each rounder than a million if you’re Indian, obviously.

So it’s definitely kind of bad. But I will point you to the highly composite numbers for a spark of hope. 24, 60, 360, 5040? Those are some nice numbers, these hit a bunch of the numbers that stand out as round to me for reasons other than playing nice with base ten. If you take stuff like the highly composite numbers, some numbers like 196 (¾ of 256) that are round in base two¹³, numbers that are round in base ten… you probably want to somehow factor in an idea of what numbers are close to being highly composite, if you don’t want to miss, say, 72, but you’ve captured most of my intuition about what numbers are round, I think. Maybe 144 and 1728¹⁴ deserve a mention? It feels way too high-dimensional to smoosh into a single scalar quantifying the roundness, though.

Beyond that, of course, is stuff like me thinking that 216, 1296, and 7776 are especially round, which I am entirely happy to grant is just me being idiosyncratic. The question of which transcendentals, complex numbers, transfinite cardinals or ordinals, and polynomials are “round” is left as an exercise for the reader.

1. I watched a video of his on dwarf planets a few weeks ago and enjoyed it.

2. One might ask how many zeroes π ends with—on the one hand, I have no idea, I don’t think this is well-defined. On the other hand, in base 2 it’s definitely “one fewer than τ does.”

3. I guess maybe prime powers can work okay but it’s going to just be the same thing as the prime except you divide everything by something.

4. Or, as with the supernaturals, maybe it’s the product of p^∞ for all primes p.

5. The pun wasn’t originally on purpose, but now I’m going to refactor the post to include it as much as possible.

6. Maybe if I’d had to worry about the cutoff between a C and a D more often, 76 would stand out as round to me too.

7. No, I didn't mean arc seconds or arc minutes, although I suppose it works for them too.

8. Well, okay, it’s 1⅔ hour, could be a lot worse. But compared to like, 100 meters? Horrible.

9. Or a time of half-day, if you’re weird and use the AM/PM thing.

For that matter, why are you telling the time of day using a system where, for one digit, you write “AM” instead of 0 and “PM” instead of 1, and also you put that digit at the end even though it’s the most significant one? This is so much worse than the whole little-endian thing, you twelve-hour people should start writing your times with AM and PM at the start.

But little-endian is better because n‑bit integers are actually n‑bit−precision 2‑adics, wherein the 2⁰s place is the most significant digit!

Well, you’re getting a little esoteric with that reasoning, but yes, it would plausibly make sense to index integers so that the zeroth bit is the 2⁰s place, the first bit is the 2¹s place, and so on. What I find harder to stomach is the practice of indexing integers in a mixed-radix system where the zeroth byte is the 256⁰s place, the first byte is the 256¹s place, but then within each byte having the zeroth nybble represent the 16⁰s place and the first nybble represent the 16¹ place. It’s the back-and-forth that’s the issue! Commit to one.

One might contest that the sign is the most significant bit, because it matters a lot whether the sign of an integer is positive or negative. What needs to be understood is that the sign bit is just the -2^(n-1)s place, which doesn’t even make it mixed radix because -2^(n-1)s ≡ 2*2^(n-2). It’s no different from any other bit.

10. Okay, maybe you need to be a very specific sort of person to ever actually confuse yourself over this. If you want to work in a large base system this is exactly the correct thing to do.

11. From the left, one indexed.

12. It will get more annoying when you add months, because then a one in the months place represents 31 days, but a two only represents an additional 28 days, except that when the decades place is 2, 4, 6, or 8 and the years place is 0, 4, or 8; the decades place is 0 and the years place is 4 or 8; the decades place is odd and the years place is 2 or 6; or the year and decade places are 0 and either the millennia place is even and the centuries place is 0, 4, or 8 or the millennia place is odd and the centuries place is 2 or 6, in which case it’s 29. That isn’t the same as 31, so the amount of days in a month isn't even consistent.

This is why the number 146097—as appeared in yesterday’s blog post—is, surprisingly, round. For 146097 is the number of days in four centuries, and once you look past fortnights you need to wait for durations that are multiples of four centuries before they take up a consistent number of days.

13. These have the “some powers of two are rounder than others” thing. 256, 1024, 65536, and 4294967296 are unusually round. Although 8192 is a round number if you’re counting bits, because that’s 1024 bytes or one kibibyte.

…Wow, I've got to learn to not make these articles 3600 words long.

14. Somehow I published the first version of this post with that saying “1792”.