100,000,000 Followers and Counting · 12:09am Feb 14th, 2015
Today Horse Voice became my 100,000,000th follower. Thank you and welcome. And I was able to (almost) reciprocate the gesture and became his/her follower number 1,000,000,001.
Counting, of course, in binary (256 and 513 in decimal notation).
Our decimal counting system is based on powers of ten, for the obvious reason that ape-descended life forms have ten digits available on both hands. It is a popular game among sci-fi fans to invent counting systems used by aliens with different anatomy.
So what base do ponies use to count? There has been much discussion on forums about this, with the most popular suggestion being base 4—one for each hoof. Seems sensible. But what can learn from the show?
The usual numerals appear fairly frequently, which does point to a decimal system, but that’s hardly a mathematical proof.
There are some anomalies. For example, it seems Rainbow Dash has a 20% larger tally than everypony else.
Here Twilight appears to be counting in base 5. Is the bottom row for Rainbow Dash?
Put 2 and 2 and 2 together. Pinkie Pie counts in base 6…
… and along the imaginary axis of the complex plane.
There is, however, a robust demonstration that ponies use base-10 from Twilight’s calculation of the square root of 546. She is correct – but only in base 10.
So how do ponies count to ten? We can speculate that the counting system included hooves, eyes, ears, tail and mane.
That makes me wonder if the different tribes originally had different numerical systems, given their different numbers of physical extremities. Pegasi add two wings, unicorns add one horn.
Maybe ten turned out to be the diplomatic option
I wouldn't be surprised at all if ponies were fluent in multiple bases -- people can be, after all. Considering how readily they use items clearly designed for humans, I imagine base 10 dominates their written calculations because they somehow just happened to wind up with ten numerals, in much the same way that they somehow just happened to wind up with tools that have handles.
Twi appears to be using an abacus designed to count in base 8 instead of base 10. A base 10 abacus has either 5 or 6 slides per row. But we see Time Turner holding up an 8. Therefore, earth ponies count in base 10, while unicorns count in the more-sensible base 8.
It's a personal headcanon of mine, that ponies borrowed heavily from us humans. Now, hear me out here. We know that Starswirl and Celestia made many trips through many magic mirrors to many different worlds, and brought back crates of information and ideas from those worlds. This likely included math concepts.
Alternatively, for those who enjoyed G1 pony, Megan and her siblings brought various information and concepts with them when they visited Ponyland, many thousands of years prior to the events of Hearths Warming.
OR... The world that ponies inhabit is really Earth, and ponies are simply the inheritors of the planet after humanity died out due to nuclear holocaust/alien invasion/plague/insert-your-own-disaster-here.
I don't know, I view it as a translation aid for the audience, kind of like how ponies speak English.
My thought is base 12.
My theory is that they used to use base 4, since they have four hooves, but as they began working with larger numbers, that system became untenable, and they switched to twelve. counting one toe and two heels on each hoof.
Granted, canon doesn't entirely back up this theory, but I'm willing to accept 'translation convenience'.
you can get base five from four hooves and a tail. Unicorns wouldn't use their horns for the same reason we ussually don't use our noses. Alternately it might actually be easier to tick off things using two front hooves and two ears instead, since they're suffecientally mobile, even if you can't see them. Pegasi could have come up with something else because of the wings, but use the earth and/or unicorn base number system. Of course getting to ten seem odd except ponies are very much not loners, so it's 10 because it's the hooves and tail of the pony next to you as well.
Or the minotaurs invented numbers and all the other races use a version of the original system, fingers or no fingers.
Complex math was first used by the minotaurs. It's why they have higher levels of technology. Iron Will and his goats had electronic headsets after all. So they've got ten fingers and ponies just use that system.
I imagine a stifling intellectual property right system led ponies to develop many different counting systems. Everypony counts in whatever system they've paid for. Rebellious souls, or those with no alternatives, develop new bases of their own....
This is a story about a pony trapped in a system so nightmarish and Kafkaesque it could only have been dreamed up by lawyers. One day she forgets to pay her counting bill. When she tries to pay the fee, she finds that she no longer has the right count. She can't pay her bill until she knows how to count, but she won't be able to count until she can pay her bill. This is known as...Catch-22.
Someone write this....
Lauren Faust dropped the ball in using decimal. If I could choose a base for the ponies, I would choose either Balanced Ternary or Sexagesimal:
Balanced Ternary is the most efficient base (radix-Economy which is the number of digits necessary to represent a number times the length of the number which reaches a minimum at e). If the ponies use an unbalanced base Sexageismal stands out because 60 is both highly composite and the least common multiple of all natural numbers upto 6, so it has lots of factors and divides things evenly into small groups of ponies up to the Mane 6. For currency, we can use Radix-Economy again:
Most nations use decimal currency. Since having to handle lots of monetary units for every transaction is inconvenient, we use intermediate denominations like thus:
000.01
000.02
000.05
000.10
000.25
000.50
001.00
002.00
005.00
010.00
025.00
050.00
100.00
As one can see, one uses powers of 2 for generating the intermediate denominations. It would more efficient to use powers of 3 because the number of coins or whatever one uses in a transaction time the number of denominations is lower:
000.01
000.03
000.10
000.30
001.00
003.00
010.00
030.00
100.00
Let us apply this to Balanced Ternary and Sexagesimal:
Balanced Ternary works perfectly with powers of 3.
Sexagesimal works poorly with powers 3:
01
03
10
20
60
It seems to work well going up, but let us generate demoninations going down:
60
20
07
02
01
¡We have a totally different sequence!
The following section contains an error. 2 and 4 tie for Radix-Economy. I shall put the error in quote-text and then give the correction beneath the quoted text:
After 3, 4 and 2 tie for lowest Radix-Economy. Then comes 5, 6, 7, 8, 9, et cetera. This gives us the equally valid sequence of:
01
04
15
60
One might as well us 4 in Sexagesimal so that one does not have to mint as many denominations, thus generating coins in these decimal equivalents:
000.01
000.04
000.15
001.00
004.00
015.00
100.00
Lauren Faust missed an opportunity for making the ponies more alien. Imagine something costing #0^ and the dialogue stating that it costs 8 bits. It would be fun to figure out that the ponies use Balanced Ternary with ^=-1 0=0, and #=+1. These are the closest ASCII-Characters to what I would like to use:
Upward-Pointing Triangle:
-1
Circle:
0
Square:
+1
I would also give the ponies a different system of timekeeping, but you already know that.
2794928
oh my god
she won't even be able to name the catch, which is exactly the sort of problem a catch-22 gives her by denying her accurate knowledge
that's brilliant, you're amazing, write this immediately
(also economists > lawyers)
Yeah, pony number systems are weird. It's possible that they inherited the decimal system along with the numerals from another culture. Not the Saddle Arabians; they seem even less likely to use base-10. The minotaurs, maybe? Only four digits on each hand, but if they count their horns as well...
2795239
But it is in the nature of economics to become friends with the lawyers....
I've also always assumed that this was likely a case of Translation Convention - while it would make more sense for them to use a base other than ten, that would be pretty confusing to young kids. There are plausible reasons why they might still use base ten anyhow, but I don't think they seem as likely.
One idea that would be interesting is if the base is tied to the average subitizing range. Unfortunately, a cursory search doesn't turn up anything conclusive (at least that I spotted...), though some bits do seem to suggest a connection might exist. Hardly surprising, but it'd be nice to have some solid evidence to work off of; as is, I've had to make do with characters just speculating on the matter. (In a different setting, but it'd apply to MLP too.)
2795054
I don't think base three, base sixty, or balanced ternary would be plausible. While they have some strong points, remember that radix economy and factorability aren't the only important things to look for in a general-use base. No living creature we know of functions in the same way our computers do, so you have to remember to account for that.
Base three is problematic because it's much easier to work with a few extra symbols than with longer numbers. Even though base ten uses three times as many numbers while only halving the average digit count, it's still much easier to use for most typical tasks. (And yes, that goes for other similar-sized bases too.) A computer would benefit from the slightly improved radix economy of base three, but living beings as we know them very much wouldn't.
Base sixty suffers from the opposite problem - using too many symbols leads to diminishing returns, and eventually slows you down as you have to sort through all of them. It's obviously doable, and I think it might still be better than base three once you've got it down, but it's not as practical as a mid-range base.
(And no, using compound bases to make a larger base doesn't count: it's functionally equivalent to using the larger of the subbases in an artifically-restricted fashion. If you disagree, please tell me how you go about writing time - I'd be interested to see the system you use.)
Balanced ternary suffers from the same issues as base three, plus the problem of being completely contrary to the seeming "default" modes of thought. While it makes sense enough once you've gotten used to it, I've never heard of any culture that uses a similar method as their normal counting system; apparently some cultures used devices with partial similarities for calculation, but so far as I know nothing that would truly count as a "balanced base" equivalent. I wouldn't deem widespread use of a balanced base in-story to be plausible unless I saw a real-world example first, and even then I'd like an explanation for why the characters' civilization chose that when most or all major real-world civilizations use "conventional" systems.
(+ 2794217)
Ultimately, for life as we know it bases in the middle of the range seem to be the best choice, at least outside of specialized applications. In regards to MLP, I'd say that by the present day base twelve seems the most likely, since it's the most composite number in the practical range and it happens to align well with multiple modes of counting they might use (four legs, four legs + horn + tail, four legs + two wings, four legs with three parts each, four hooves with three parts each, etc etc etc - and that's not even considering other species). If, as seems likely, they started out with different bases in common use, base twelve would be the obvious compromise.
2799164
As you know, the Babylonians used Sexagesimal, so it is a plausible base for ponies.
I have a plausible path to Balanced Ternary for ponies:
Ponies start counting using their forehooves (Base2). The Ponies discover the concept of 0. They discover the concept of negative numbers. A pony-mathematicians discovers that if negative -1 has its own symbol and ponies switch to Base3, the mathematics gets much simpler because multiplication is still easy and carries occur only 2/9ths of the time:
* As you know, the carry-rate for addition in Binary is ¼ (it is 2/5ths for decimal). The carry-rate is only 2/9th for Balanced Ternary.
* As you know, in Binary, , when one multiples 2 numbers, one can replace every 1 in one number with the whole number of the other number, shift the radix-point appropriately, and add the partial sums to get the final answer:
11010001*00100100=
0001101000100
+
1101000100000
=
1110101100100
In Balanced Ternary let:
▲:
-1
◯:
0
⬛︎:
+1
The rules for multiplication are the same except one switches the sign of digits when the digit is negative:
▲⬛︎▲*◯⬛︎⬛︎=
0▲⬛︎▲
+
▲⬛︎▲◯
=
▲◯◯▲
We get negative -28.
¡The ponies see the advantages of Balanced Ternary and switch from Binary to Balanced Ternary!
I believe that this is a plausible path to Balanced Ternary.
In Balanced Ternary, truncation is the same as rounding:
Let us suppose that one calculates something involving circles. The 1 true CircleConstant is τ (Tau):
c / r = τ ≈ (Base10: 6.283185307179586) ≈ (BaseBalanced3: ⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯;◯▲⬛︎,⬛︎◯▲,▲◯▲;⬛︎▲▲,◯◯◯,◯◯⬛︎
⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯;◯▲⬛︎,⬛︎◯▲,▲◯▲;⬛︎▲▲,◯◯◯,◯◯⬛︎ is a bit much, so since truncation and rounding are the same, we can chop off extra digits like thus:
⬛︎▲◯.⬛︎◯▲,▲◯▲,⬛︎⬛︎◯
Post Scriptum:
¡I got to play with my dice for generating the digits! This reminds me about a joke:
A D&D-Player is in Geometry-Class. The Teacher asks the D&D-Player to name the 5 Platonic Solids. The D&D-Player responds thus:
2799164
This BlogPost and my interactions with MSPiper inspire me to write a BlogPost about why Ponies would used Balanced Ternary:
¡A Plausible Way For Ponies To Develop Using Balanced Ternary As A Base!
2803752
Looks interesting, but I'll have to wait on that til I have more time to think about and respond. One immediate correction, though: the name's based on my initials, not Ms. - Piper's the last name. I suppose it's an understandable interpretation, but it's surprisingly the first time I've seen it.
2804071
I shall collapse it down to the string “MSPiper”. I hope that you like the BlogPost you and Pineta helped inspired. I also caught an error in the reply about Radix-Economy and Sexagesimal on this BlogPost:
I forgot that 2 and 4 have the same Radix-Economy, so one can use either powers of 2 or 4 for generating Denominations of Sexagesimal Currency.
Ah, yes, I had forgotten the square root. That's one more item for the list of numbers that would be obviously wrong if not translated.
You're missing the other side of the argument here, though, which is the question of whether the numbers actually are being translated to their base-10 representations for us. And to answer that, I point to the sorts of numbers that are chosen under the sorts of circumstances where one chooses round numbers: 1000 years of banishment, 100 moons between Apple family reunions, 30 moons between openings of the mirror. I won't entirely discount the possibility of ponies using base 2 or 5, but this does strongly suggest base 10.
2924603
And exactly what '30 moons' means is up to Princess Luna...