What's a good numerical base?

[Thue]

Of course they can't. In my example above, see how 20/3 gives rise to 6,666... . Stupid North Americans would probably just keep doing

20/3 = 6 + 2/3 = 6 + 0.1*20/3 = 6 + 0.1*6 + 0.1*2/3 = ...

And potentially waste lots of time. It is for your benefit that I am suggesting a base which includes a factor 3, though you don't seem to be grateful .

[onodera]

Why have a base-based system at all?
Let them have separate names for the first 136-152 numbers, and nothing after that. Larger number names are different between the colonies, so they are fuzzy somewhat.
Of course, this will hinder the development of math, but what if these funghi suck at manipulating abstractions? They don't need to talk about millions, they just say "very many spores".

[Lul Thyme]

Originally posted by Asher

Fact: Thue is a European and doesn't understand that people in North American can divide by 3 without resorting to a new number system.

I don't know why you're still arguing this.

There are obvious advantages to a base-12 system (or, in general, to bases with lots of divisors), some of which Thue has mentioned.

In fact, behind base 10 (and base 20 to a certain extent), which are due to historical accident,

base 12 (which is the smallest number divisible by all natural numbers up to 4) and other high number of divisor bases (like 60, which is the smallest number divisible by all natural numbers up to 6) were the most frequently used, until the digital age. Basically, base 12 was the most useful for what people needed to do throughout history, until recently.

I'm not actually arguing that base-12 is "better" than any other one, since this depends of course on what you want in your number base. You're of course right that if you're using binary-based computers, then a power of-2 base makes sense, but that's hardly relevant to the opening post.

Even your base-16 is not really an "obvious" choice among power of 2 bases. Why not 8, for example? It comes down to a number of different digits vs length of numbers.

To answer another comment of yours, yes 3 is a "common" number. In fact, smaller numbers are generally more "common" than larger ones, in a vague sense, but for almost any reasonable usage of "common".



Anyway, to reply to the original post, historically, people have found it advantageous to have bases with many small divisors
For one thing "round numbers" like 100, also inherit this property and if you're going to use your base to go between units, it makes it easier to divide units in other units by small (and hence common) integers.
For example, we talk of "percentage" because a 100 is a power of 10. If we had base 12, we might talk of "144age". In 144age, a third becomes nicely expressible, which isn't true in percentage (although the reverse is true for 5, as Asher so wisely pointed out, but 12 has relatively more divisors than 10).

If you dollar is 144 cents, then a third of a dollar is an integer amount of cents (but a fifth, etc...).

You can also make arguments for a bunch of other bases.
For example, there is a trick to see if a number written in base-b is divisible by divisors of (b-1). For example, to see if a number in decimal is divisible by 1,3 or 9, you can just add its digits and see if that is divisible by 1,3 or 9 (and do this recursively if necessary). So, if we used base-13, for example, we could do the same trick for 1,2,3,4,6,12.

I'm not saying that this is an important reason to choose such a base, just pointing out that it really depends what you want to accomplish.

What happened on Earth is that we mostly chose a system independently of its abstract properties and then just stuck with it.

If you are going for realism, maybe that is the way to go.

[rah]

Gee, I'll bet that even Alinestra Covelia has stopped reading this thread.

[Lul Thyme]

To mention another pre-digital yet not necessarily ancient example : the unit systems.

While of course metric is generally superior to things like the Imperial system, that's because the base is at least consistent and the units are related to each other in simpler ways.

Throughout history, unit systems used numbers like 12 a lot (12 inches to a foot, 24 hours in a day, etc...) because it has more factors and hence you can divide a foot by more numbers to get an integer number of inches.

As I said earlier, the metric being generally superior is in fact NOT related to the above property, and we could have a system with all the advantages of the metric in base-12 (if we used base-12 in our usual numbering system).

I want to emphasize, again, that I'm not even arguing that base-12 is better to anything but simply pointing out that there ARE advantages to "high number of divisor" bases.

[Q Classic]

There's also how they'd break magnitudes into groups. Would they be like the West, focused on orders of 3s (100 000, 10 000, 1 000) (100 10 1)?

Or like the East, in orders of 5 (1 000 000 000, 100 000 000, 10 000 000, 1 000 000, 100 000) (10 000, 1 000, 100, 10, 1)?

Would their language be more akin to the long scale (1.0e9 = thousand million), instead of the short scale (1.0e9 = billion)?

[Q Classic]

The problem with this discussion is much of it is focused on what is the best system, not what would have developed organically. Base 10 arose from the digits on the hand; base 12, perhaps from the digits on the hand plus the fists; base 20, digits on the hands and feet.

Base 11, maybe from the digits on the hand plus the head.

How would this species have come up with a specific base system if they don't have a constant number of limbs or digits? I'd wager that they'd develop it by external physical features--hence, my suggestion of the number of moons, or some such. Is there something constant about their physiology? Perhaps based on the number of eyes; or the number of mouths.

Overall, though, I'm curious how your friend is going to introduce this into the story; as Asimov put it bluntly in his novel Nightfall, itself an extension of a short story he once wrote--sometimes, to make the story more accessible, it's perfectly reasonable to draw analogues when possible. Instead of having a character attracted to another's cavernous gleepmorp, have them fascinated by the other's strong jaw.

Now, if the number system isn't going to be front and center, then there's no need to explain exactly why they use a base 24 system or what not; it's something that can be written into the details, so that the numerology can be picked up by careful readers, but for those who don't care, it doesn't prove a stumbling block.

[Alinestra Covelia]

My friend wants to show that a staggering amount of time has passed, and that the mushrooms can know this.

He originally thought of them as having three fingers, and three arms, and he used the term "fulfilled threes" (a tetration of 3, in this case 3 ^ (3^3)) to describe the amount of heartbeats that have gone by since the mushrooms' shared consciousness first achieved sentience.

Back deducing from this, it indicates the mushroom life form has been around for at least 241,000 years or so.

[Alinestra Covelia]

Mrs Snuggles makes a good point in that the mushrooms don't have a convenient constant number of limbs or digits to use as their numerical inspiration.

But consider then that maybe they would be freed from the mental inclination to use a less-than-ideal base, and could instead sit down (or in the case of mushrooms, stand still) and think about what numbers would work best.

I find the 16 base argument to be quite persuasive, if only because of the issue of it being easiest to halve things by eye.

The base 12 system appeals to me because of its decent number of common denominators. Bing however has said if we like base 12, we may as well go base 6.

The mushrooms merely prefer to have three legs so they don't fall over on rocky terrain. There's no requirement that they have three hands and three fingers on each. In fact, part of the story is that they intentionally sprout two arms and five digits per hand and try to lumber about on two legs (clumsily) in order to put humans at ease during first contact.

[BeBMan]

Originally posted by Alinestra Covelia


The mushrooms merely prefer to have three legs so they don't fall over on rocky terrain. There's no requirement that they have three hands and three fingers on each. In fact, part of the story is that they intentionally sprout two arms and five digits per hand and try to lumber about on two legs (clumsily) in order to put humans at ease during first contact.

Damn cultural relativists!

[Asher]

Originally posted by rah
Gee, I'll bet that even Alinestra Covelia has stopped reading this thread.

I got really bored also especially when that guy from Slovenia or whatever responded. Cripes he's boring.

And he didn't once attack the basis of my argument, no doubt because it was rock solid -- that Thue is European and can't do math.

[snoopy369]

It's important to realize that while, NOW, we know that 16 is probably the most useful to us (in a digital age and all), the folks creating this system may not have been thinking of digital, or boolean, or anything like that; they were thinking of counting things. Thinking of how it might have come about is very important, and I suspect (as Lul explains) 12 is the most likely to have come about in this manner (or, perhaps, 60 ?).

I do like the suggestion for different groups having different bases, though - that's a good one. You could have the Asherites with base 8 (16 is probably too many digits), and the Thuites with base 6 or 12, and have them spar regularly

[Theben]

Over an apparently worthless frozen island.

[carnide_]

As important as the finger business, are the perceptions skills. How do they discriminate symbols? How can they communicate how to behave?
Dont forget that two of them, given the same expression, should manipulate it at will, but arrive at the very same result.
As number crunching math may be, it is also a behaviour description language.

If their symbols are chemical based, not visual ones, a base 2 or even base 4, DNA like, should do the trick.

[-Jrabbit]

Three is primal to these fungi. It seems wrong that their base counting system would not accommodate it. So a simpler solution is best.

Two hands, 5 fingers = base ten.
Three stalks, 2 pseudopods = base six.

Done.