Why do humans count in base ten? Because while each of our four appendages end in five digits, we walk on two of those and tend to cover them up a lot. So when the idea of counting first started clicking for whatever humans (or pre-humans) first came up with it, they compared objects to their fingers. "I've got a hand of good flints, he's got more than a hand, together we should be able to make two hands of spearheads, maybe more."

Larry Niven's Kzin race counts in base eight. The Kzinti are felinoid and have only four digits on each limb. The number 32 is, by our way of thinking, three tens and two ones; to a kzin, it would be four eights. (Similarly, Niven and Pournelle's Fithp are an elephant-like race with branching trunks that end in a total of eight prehensile splits; they also count in base eight.)

With very few exceptions, the standard in science fiction -- if the question is addressed at all -- is for a race's counting and therefore all of their mathematics to be based on their fingers. (One exception is Babylon 5's human-like Minbari race, which figures in base 11. They count based on their ten fingers and the top of their heads, which may be related to the impressive bone crests they grow up there.) But what do you do with a race that has no fingers and no manipulatory limbs?

Okay, technically, ponies have a total of four digits -- one on the end of each limb. And if they were bipedal in nature, I might think they were naturally inclined to binary math, counting with just those upper limbs. But being quadrupeds, I think they probably do their math in base four. Need to count something? Stomp, stomp, stomp, stomp! Use a different hoof each time to keep it straight, and from there it's a short leap to larger amounts, and not a long journey to even more complicated math.

If that's the case, then just as we have words for ten squared (a hundred) and ten cubed (a thousand), the pony language would have words for four squared, four cubed, and so on. And to use my example of 32 from before, to a pony that would be two <four squared>.

3276659

How about multiples of five? Left fore hoof, right fore hoof, left hind hoof, right hind hoof, tail. They can do a lot with their tails, from what we've seen on the show...

--arcum42

3276659

Have you read Twilight makes first contact? In that story ponies are using a base of four, which I think fits the best.

3276659
3276671
Actually, this could go both ways. If ponies counted in base 4, why wouldn't humans count in base 2? Sure; vast majority of humans have 10 fingers and all, but would that be what anchors the default base system? Another system, that comes to mind is also really practical, and it's in widespread usage, especially since the increasing usage of internet: Hexadecimal.

But here is a thing:
From the transcript of Equestria Games:

Spike: Aw, she's just nervous, that's all. Perfectly understandable. Whenever I'm afraid I'll forget something or start to panic, I have a simple trick. I count to ten, and by the time I'm done, I've calmed myself enough to get the job done right every time. Easy-peasy, cider-squeezy!

It seems, that Equestria did adopt the base 10 counting mechanism by default, just like us. Further indicator is the transcript from Leap of Faith:

Granny Smith: Oh, quit your fussin'! I had enough tonic to do a dive ten times as high!
Flim: Twenty times, by my count.
Flam: Thirty, with a favorable breeze.

All jumps are by the base 10.

But, I have a better question:
Since base 10 is established default, what would any other base system look like? Would a system like hexadecimal be even acceptable?

~Twi

3276687

I'd count that as part of transliteration into common English phases. Clearly Spike would want to use the ancient alicorn and dragon counting system: Octal. (4 limbs, tail, two wings, horn / two hands, four fingers each.) Unicorns and pegasi may have used base 6 and 7 at one point or another as well, but it'd make sense to standardize on base five once the three tribes got together.

--arcum42

3276689

part of transliteration into common English

I can accept that. But my guess would be that before standardization and mixing of the three races it would be much more rational for each race to have its own system:
Draconic: Base 8
Pegasi: Base 7 (4 hooves, 2 wings and a tail)
Unicorn: Base 6 (4 hooves + horn + tail)
Earth pony: Base 5 (4 hooves + tail)
...and so on.

We need to understand, that before the events of Hearth's Warming Night, the different types of ponies could be treated as completely separate species rather than types of the same animal.

Edit: Extra thingy... If they standardize the base counting system; why would they settle with base 8 and not base 10?
~Twi

3276687
Why fingers were used as a base for humans was already explained above. Why Ponies wouldn't be able to use that base, not having digits, was also explained.

Als0, Hexidecimal (base 16) is used simply because you can represent a byte as a single character, rather than the 8 characters it takes to represent it in binary. A single byte can represent 256 different things when related to a key, such as ASCII. Having one character able to hold that much variety of information is incredibly useful.

3276659
Among ponies, it seems to me pegasi or unicorns would be the first to invent math. Pegasi were the warriors, and being able to conceptualize numbers of enemies/allies and calculate odds is a huge advantage, and little motivates innovation better than life or death situations such as the battlefield. I'd imagine they might use a systtem based on the number of primary feathers on their wings, as those are often "handy" for them.

Unicorns would either quickly pick up on the numbers from pegasi, or possibly be intellectual enough t have thought it up first. In that case, I am not too sure they would base their number system off of their own body. A base 4 system isn't to bad, but You do very quickly rise in digits under it. They might base their system off of celestial events, such as lunar phases or the year. Before Discrd's reign, Unicornia was responsible for the sun and the moon.

It may also be possible that Celestia and Luna, after rescuing the ponies from Discord, may have implimented their own system to act as a further unification. Something like base 12 (four legs that all ponies have and three tribes of ponies) might be the sort of thing they'd come up with. B12 would be a pretty nice system, I feel.

3276697

Right. And then all came together and mixed, and the lesser used bases would have fallen into disuse. My guess would be that 6 and 7 fell into disuse, with remnants left in pegasi and unicorn culture and tradition. Five would become the most likely common base, due to all three races having four hooves and a tail. I'd like to think that octal gets used in government and mathematics, with Celestia having been raised on it. (That and I like to give the dragons some culture. I like thinking of ancient dragon mathematicians counting the coins in their horde...)

Also, octal is actually easier to divide and multiply with then decimal, and is probably the next most common base after decimal, hexidecimal, and binary in our world.

--arcum42

3276706
I see. It makes sense. Although; I'd like to think that any multiple of 4 would be a rational base system... Both, base 4 and 8 all come handy, since it's also compatible with dragons. Although, I'm interested... Does Equestria accommodate enough dragons to warrant the compatibility, or was their counting system rational enough to be adopted by ponies?

~Twi

3276717

Part of this is my biases due to reading a lot of fantasy, but while teenage dragons aren't much to look at, dragons seem to be near immortal, and I tend to see there being a fair amount of ancient dragon scholars. I'm not sure how plentiful they are, but keep in mind we only see one corner of the world, and the section specifically dominated by ponies. There may be dragons in plentitude elsewhere.

I also tend to see there as having been more alicorns at some point in the past, and they'd have been using octal as well. I've noted that Equestria seems to have more than one writing system as well, which is one reason why I see no issue with multiple bases being used as well.

--arcum42

3276723
You know; I've been wondering about that as well... Just how many things did ponies learn from dragons? And what did they adapted from them? Was writing one of them?

~Twi

3276731

Yeah, we've been shortchanged a bit on what we've seen of dragons. The dragon teenagers were just a bunch of brats hanging around making trouble, and the dragon Fluttershy tackled didn't really seem that smart, though he had been sleeping. There was another, but that was such a horrible episode that my minds blanked it out.

As far as Spike himself, he may seem clumsy and the butt of every joke, but he is a baby dragon. By dragon standards, he's probably barely out of diapers, and will probably be true until he's older than Granny Smith. One day he'll likely be a very old and wise dragon.

It would be interesting to see how much comes from dragon sources originally. It's an area I'd like to see explored better.

(And now I've got My Little Dragon and The Ambassador's Son on my mind...)

--arcum42

First off, I'm pleasantly surprised by the participation in my nerdy topic. Thanks, all!

3276671
Also plausible.

3276682
I haven't, but I may need to check it out.

3276687
Obviously the dialogue on the show is translated for the benefit of human viewers.

3276703
Thanks for backing me up.

And it's entirely possible that the pegasus and unicorn tribes came up with math separately from each other. You're absolutely right that knowing numbers would give you an advantage in battle, and I'm also inclined to think that unicorns have a little bit of an edge in smarts as well as magic.

3276737

One day he'll likely be a very old and wise dragon.

I agree with that. As much as he's used as a punching bag in the episodes, he does live with Twilight and treats her as his big sister / motherly figure.

And even for being a baby dragon, he does seem to be very smart...

There should be more stories about grown up Spike. Unfortunately, in most stories featuring more grown up Spike he's just used as a sex object or a romance device.

I've began to hate that.
~Twi

3276659
I've wondered about that, myself. In my headcanon, they got their numbering system from another culture, probably the minotaurs.

Of course the minotaurs as drawn on-screen probably don't have fine fingers per hand, so that's pretty questionable even to me.

One of our senses deals with the physical position of our limbs, which allows us to count on our fingers without actually looking at them (among other things). This gives us 10 boolean values which we can abstract as an unsigned counter.

So what if ponies don't count on their hooves? They could count with flicks of their tail - which still uses a motor neuron feedback to help them keep track of counting - but stores it as a stream rather than a set of parallel booleans.

So how would one remember a number as a series of time-separated ticks? Hard to say. It may remain an interminable mystery, much like how I manage to find my way from my desk to the break room on a regular basis without getting lost.

Let's face it. We're terrible with numbers. I mean, we're pretty good with anything in the single digits, but beyond that, it starts to blur a bit.

The Trolls of Discworld numerical system goes something like this:
One.
Two.
Many.
LOTS!

The alphabet of numbers gives us some handle on dealing with values larger than this. It's full of a lot of nice shortcuts, like counting by tens, buying RAM in exponents of base-2, knowing that the constant of cosmological expansion tends towards "1" while your bank account tends towards "0", or knowing that there are 6.02E23 atoms in a gram of hydrogen, and about 1.0E82 electrons in the observable universe.

Without paper or computers, humans are still pretty bad at numbers. Unless they play ridiculous amounts of tabletop games. We are bad at math in the same way we are bad at punching trees until they fall over. Therefore, to appease the gods of laziness, we invented ways of dealing with math and tree stumps that are far more effective than we could ever dream to be on our own. However, this overlooks the fact that while we are bad at math, we are frighteningly good at applied physics.

This is how our brain works. A lot of dedicated circuits to solve complex equations - except without any numbers on the dials. Those, we had to make up.

"The simplest thought like the concept of the number one has an elaborate logical underpinning"
- Charles Barkley

3276793

A baby dragon is relative. I tend to peg him as being roughly the same age as the cmc, though he may even be younger than that, thinking about it.

I get very tired of him being used as a walking pratfall and punching bag on the show, really. And I agree about older spike in fanfics. Spike has a tendency to be forgotten in a lot of fanfics, too. Not to mention episodes. So Spike wouldn't like to make rock candy, or to go to Twilight's birthday party?

--arcum42

They use the Natural Logarithm as their base.

3276659
Not all human civilizations use base 10 counting system. For example, many ancient civilizations used base 60 counting systems (in fact, our method of keeping time comes from a base 60 system since we have 60 seconds in a minute and 60 minutes in an hour). Base 60 is quite convenient because it is a highly composite number and thus has many divisors. So perhaps ponies would chose their number system logically and use a highly composite number like 6 or 12 as the base.

3277164
I briefly hinted at some intellectual/cultural exchanges with minotaurs in one of my stories, but I had it happening long after basic stuff like math.

3277200
Bonus points for a Terry Pratchett reference.

3277827
The Phoenicians were oddly fond of the number 60, sure, but they wouldn't have arrived at destination without at least one or two stops along the way...

3276659
I figured base 10 could work with ear positions—five with the left and five with the right. Straight up, forward, out to the side, all the way down, and all the way back.

3279017
It could be done, sure -- just like I can count to 31 on the fingers of one hand -- but is it intuitive? Not like counting hooves would be. (Or hooves and a tail, or hooves and wings, or other examples in the thread.)

3279019

Well, they count hooves in canon...

[youtube=lw7eDGe4yP0]

--arcum42

3279021
There we go -- ponies count in base six. My mistake. Thank you, Pinkie Pie!

3279019
Considering they're about as prehensile as fingers and their positions are an indispensable part of expressions, I could see it being pretty intuitive, yeah.

3279034
Development of communication always starts out simply. One finger (or hoof) equals one object, two fingers (or hooves) equals two objects, and so on. I can count in binary on my fingers, but I don't know of any civilization in human history that adopted binary from the get-go. I could see ear positions being used for more nuanced, detailed communication (in addition to the obvious emotional indicators that they already are), but not as a basis for foundational math.

3279037

I suppose binary wouldn't necessarily be that difficult for ponies. You could do it with hoof cadences, with clip-clop-clip-clop-clip-clop being 42, for example...

--arcum42

3279037
Maybe I'm not explaining it clearly enough, because what I'm imagining is incredibly simple, especially since in horses there are already about a half-dozen discrete ear positions already naturally used for communication. Just call them numbers instead of emotional states and double the amount by having the ears work separately.

It's not natural for us to think of ears that way, but that's because we don't gesticulate with them in conversation, which I think makes them a natural site of making up new gestures for things like counting. In the same way, they'd probably have trouble imagining how we could mentally keep track of the position of all our fingers.

3277487
That's actually not a bad idea...

We're all discussing about ponies having a base system based on a variety of natural numbers (1,2,3,4,5,6,...).
But here is a real question: what would stop them from using algorithms or complex numbers as a base counting mechanics? Why not make base negative 3.1415926 and scales logarithmic instead of linear?

And before anypony would like to attack me with a blade of "rationality" and bludgeon me with a bat of "convenience", let me explain a little something:
With exception of Martian, we all picked a base system that's comfortable for us. Whether being base 10, 5, 4, 6 or whatever. We all picked it because those are whole, natural numbers. We're so ingrained with those numbers, that they compel to us. But here's a thing: Equestria is fueled by magic. There is hardly any chance that a pony's brain works like ours. Even the earth ponies are no exception to this rule, as their connection to earth helps them grow crops and take care of environment (it would be waste of words to connect pegasi and unicorns to magic, so I won't).

As far as we know, 1 + 1 might not be 2, so all the proposed base systems collapse.

And instead of learning simple math in the first year of elementary school, the ponies could quite likely learn what we consider "higher-level math".
~Twi

3276659
Here's some damning evidence.

I'm not entirely sure how that works, but... whatever.

Just lends more credence to a base two system.

3344049
It clearly works through a generous use of FM*.

*No, not Frequency Modulation. I'm referring to "Fucking Magic".

3276687

Well the thing is that every base is base ten from that base's perspective:

3606035

Well the thing is that every base is base ten from that base's perspective:

All bases are base 10.

Except those that aren't.
~Twi

3606046

The joke is that whatever the value the base has is represented as "10" in that base.

3276659

I assume base 12 for the ponies. I figure they used to use base four (four hooves), but when they invented mathematics, numbers were too long (especially if you've got to write them out by mouth), and there weren't enough factors, so they changed to 12.

It's a natural change, since they can count heel, toe, heel on each hoof, and reach 12 that way.

Interestingly, I've heard that there were some ancient human civlizations who used base 12. They counted individual finger segments on each hand (try it--use your right index finger to count off finger segments on your left hand).

I don't know if the 12 inches in a foot or 12 hours in a day is a holdover from that, but given how screwed up the Imperial measurement system and the timekeeping systems are, I wouldn't be surprised if it was.

I figure that ponies started out with Quaternary, but soon discovered the advantages of using a highly composite number which is also the least common multiple of all natural numbers up to a point. The only not-horrendously-large base which is both the least common multiple of sequential natural numbers and is also highly composite is Sexagesimal.

By the time, ponies discovered the best base, Balanced Ternary, because of its low radix-economy (the number of digits need to represent a number times the number of numerals the base uses). Radix economy is lowest at e. Balanced Ternary uses the numerals -1, 0, and +1 Let us represent them as T, 0, and 1:

TTT = -9 -3 -1 = -13
TT0 = -9 -3 0 = -12
TT1 = -9 -3 + = -11
T0T = -9 0 -1 = -10
T00 = -9 0 0 = -9
T01 = -9 0 +1 = -8
T1T = -9 +3 -1 = -7
T10 = -9 +3 0 = -6
T11 = -9 +3 +1 = -5
0TT = 0 -3 -1 = -4
0T0 = 0 -3 0 = -3
0T1 = 0 -3 +1 = -2
00T = 0 0 -1 = -1
000 = 0 0 0 = 0
001 = 0 0 +1 = +1
01T = 0 +3 -1 = +2
010 = 0 +3 0 = +3
011 = 0 +3 +1 = +4
1TT = +9 -3 -1 = +5
1T0 = +9 -3 0 = +6
1T1 = +9 -3 +1 = +7
10T = +9 0 -1 = +8
100 = +9 0 0 = +9
101 = +9 0 +1 = +10
11T = +9 +3 -1 = +11
110 = +9 +3 0 = +12
111 = +9 +3 +1 = +13

The 1 True Circle-Constant τ (Tau) in Balanced Ternary:

τ ≈ 1T0 . 10T,T0T,110 , 0T1,10T,T0T , 1TT,000,001 ≈ 6.28318530717959

τ ≈ +9 -3 0 +⅓ 0 -1/27 -1/81 0 -1/729 +1/2187 +1/6561 0 …

Unfortunately, for inventing Balanced Ternary, ponies would have to understand both negative numbers and Radix-Economy. By that time Sexagesimal would be too deeply rooted, so they would be stuck with it, just as a bunch of apes are stuck with decimal.

3277487

e is transcendental, so integers in it would all be nonterminating radices, but you are onto something:

Radix-Economy is lowest closest to e. Radix economy is the number of digits needed to represent a number times the numerals used in the base. 3 has the lowest Radix-Economy of any IntegerBase. The best base is Balanced Ternary. I wrote a comment on this blog explaining this:

Balanced Ternary is the best base, but ponies probably use Sexagesimal anyway.

3279129

We experimented with negative, imaginary, and complex bases in computers. The advatage of negative bases is that one need not have a way of indicating sign. In the case of imaginary and comples bases, one need neither indicate sign nor whether it is real imaginary or complex. The disadvantages are that representations of something as simple as small natural number are long, with many necessary significant figures and simple arithmetic is more complex. These are the bases we tried:

-2
2i
−1±i

Certainly, one can write numbers on the imaginary plane as a simple stream of digits, but at the cost of making natural numbers require long digit-streams for representing small natural numbers. I would use Balanced Ternary instead.