Why Base 10?

We (at least the math nerds among us) are having an interesting discussion on the Wesnoth forums about why we use base 10 rather than, say, base 6 or base 12. Some interesting ideas have emerged. They all have to do in some way with what base we count in. There’s in a sort of order here, but I’m not trying to make an argument for or against something, I’m just presenting some concepts. Feel free to read some parts but not others.

First off, some notational issues
Talking about bases is hard because we’re so used to thinking in base 10. There’s no real reason for this. In fact, if we had grown up using what we currently term base 6, we would call it “base 10″, because the symbol “10” would then represent the number we currently represent by “6”. What we currently call base 10 we would call “base 14″.

Because of this, I’m not going to use the notation “base 10″, “base 12″, etc. That supposes a system based on the number that equals 5+5. But what to use instead? We have to use something. Since we want a base that presupposes nothing, the most logical choices are the unit base (base 1), and base infinity. In base 1,
1=1
2=11
3=111

…etc

But this gets very cumbersome very quickly. Base infinity, on the other hand, requires new symbols for every number. This sounds excessive, but it’s definitely possible. In this post, the sequence of symbols will be 0,1…9,A…Z,a…z, which brings me up to 6bA. That’s high enough for the purposes of this post. So I’m going to use base infinity. (Unfortunately the symbol “b” represents both that the next number is the base we’re in, and the number 37. But hopefully it won’t ever be ambiguous.)

As you may have noticed, I slipped in this thing 61bA. I’m going to use that notation to represent numbers as written in a given base. If I don’t give a base, then the number is base infinity. Obviously, I’m going to use base infinity to represent what base I’m in – that’s why I wrote bA instead of b10. b10 is meaningless – obviously whatever base you’re in can be thought of as b10, but that doesn’t give any information.

I don’t use bA especially in this post except for decimals. This is because you can’t really write decimals in base infinity, and I might as well use bA for them rather than some random base that no one will understand. However, I won’t use bA for the part of the decimal to the left of the decimal place, only for the part to the right. I hope that made sense.

Incidentally, the reason this post is marked “language” as well as “mathematics” is that the question of how to represent numbers when writing them on a page is really one of language just as much as it is a mathematical one. It has to do with how we convert information from symbols – written, spoken, whatever – to abstract ideas.

Why we currently use bA
“Because we have ten fingers”. Perhaps. But A fingers could just as easily lead to base B – we count up to A on our fingers, and then say “10!”(bB) when we put all of our fingers down again. Or we could not count the thumbs, and thus get base 8. Or go one hand at a time, and get either base 5 or base 6. And clearly this argument about it being based on the hand isn’t really applicable, because different cultures have used different bases throughout history, many of which weren’t based on the number of fingers we have:

  • base 8 (Native americans)
  • base K (Mayans)
  • base y (Babylonions)
  • base 2 (Vikings)
  • base C (Imperial- system, Nigeria, Nepal)
  • base 2 (specific Australian Aboriginal nations)
  • base As (Romans, Chinese, Hindus)

(list courtesy of appleide)

So it seems that the reason we use base A is just that Rome, which took over Europe, China, which dominated the Far East, and India, which dominated southern Asia, used it. Mathematics, like history, is written by the victor.

Advantages of using other bases
Firstly, it seems advantageous to have the base have a large number of factors. It makes both multiplication and division easier. Now, A has only two factors (other than 1 and itself) – 2 and 5. Another proposed base, C, is slightly higher, but it has four factors – 2, 3, 4 and 6. Base 6 would give us only two factors, 2 and 3, but that would be fully half of the possible factors. So either of those bases has an advantage over A in terms of factors.

An often-raised objection against base 6 is that, since 6<A, the number of digits required to show a number b6 will increase much quicker than the number of digits for the same number represented bA. This is true to an extent, but the number of extra digits isn’t really that high. Taking the multiplicative inverse of the log base A of 6 gives us
1/log6(A) = 1.285097209...
So about 4 digits b6 for every 3 digits bA. That’s not an excessive amount.

Another base that appeals to me is b8. The concept of “subitizing” is that you can almost instantly determine the size of any set of objects with up to 4 objects in it. An example – when playing cards, in a game where you have, say, 13b10 cards, the easiest way to make sure you have the right number is to look at it as 31b4 and go “1234-2234-3234-and1″. Each set of 4 is counted almost instantly. Try it – it takes must shorter than going “1-2-3-4-5-6-7-8-9-10-11-12-13″. Given that fact, b4 would sound good, except 4 is probably too small (there’s 1.66096…(bA) digits per digit bA. So b8 is the next logical choice.

Strange Base Choices
So it seems that you want bases that have many factors, are fairly low (so that you don’t have to memorize a whole bunch of symbols), but aren’t too low (so that the number of digits per number isn’t too high). But there’s no reason you have to do this. I’m using base infinity a lot in this post. You can also do things like negative bases, fractional bases, and even irrational and transcendental bases.

Consider base e, ~2.718, (as distinct from e, =40bA). In theory it sounds like a good idea because e is the natural base of logarithms. No reason to have to memorize two different bases for two different contexts. But the problem with that is…
(b10)|(be)
1=1
2=2
3=10.2817...
4=11.2817...
5=12.2817...
6=20.5634...
7=21.5634...
8=22.5634...
OR 100.6109...
9=101.6109...
A=102.6109...

I find it interesting how 8 can be written 22.5634 OR 100.6109. this is because 8 = 2*e+2.5634 but it also equals e^2 + 0.6109, both of which would be valid under be.

Another, um, interesting thing to try is a negative base. Here’s b-6 analyzed:
(bA)|(b-6)
1=1
2=2
...
5=5
6=150
7=151
...
11=155
12=140
13=141
...
35=115
36=100
37=101
...
42=250
...
216=15000

You go up two decimal places at a time, and count backwards, sort of.

Now, let’s try a base between 0 and 1. How about base 1/2?
(bA)|(b1/2)
0.25=10.0
0.50=01.0
0.75=11.0
1=0.1
2=0.01
3=.11
4=0.01

It’s essentially inverted binary. Bases from numbers 0<number<1 function like base number^-1 but written right to left instead of left to right.

Another interesting idea isn’t exactly a legal base. It involves using both positive and negative numbers. It only works with odd numbers, I think. Here’s how you do it with b3. You have three symbols, say, “!, 0, 1″, representing “-1, 0, 1″. Then numbers are written
(bA)|(odd base)
0=0
1=1
2=1Z
3=10
4=11
5=1ZZ
6=1Z0
7=1Z1
8=10Z
9=100
...etc

Any other ideas for strange ways to represent numbers?

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10 Responses to Why Base 10?

  1. Cole says:

    I like base 6 or base 12 because you can count up to 35 on your fingers in base 6 (and up to 1295 on your fingers and toes, if you have agile toes), using the right hand fingers to represent the 6^0 place and left hand fingers to represent the 6^1 place. 35 is the HIGHEST you can count on your fingers in ANY base if you only use two places, AND conveniently this lets you use each hand as a separate place, which makes it much simpler. It’s easy to convert from base 6 to base 12, and 12 has the advantage of fewer decimals needed and having an additional factor (4, which as you describe in this post, is an important one).

    Actually you can count up to 1027 on your fingers in binary but that’s confusing as hell.

    I like your argument for base 8, I hadn’t considered it before. There’s also a serious advantage to having a base that is a multiple of 2, so that doubling and halving only requires moving decimal places.

  2. Cole says:

    That last thing about a base that is a multiple of 2 was a lie. I meant a base that was a power of 2, but even then the statement doesn’t hold true.

  3. Counting with your toes would require serious dexterity. But I totally agree about counting in base 6 using one hand for the 6^0s and the other for the 6^1s.

    Also, it’s really fun watching people’s reactions as I count on my hands 1…2…3…4…5…10!…11…12… […] …54…55…100!

  4. Cole says:

    Ah but you can’t reach 100, if, like most people, you only have 15 fingers. The highest you can get is 55.

    I’m not sure how I feel about counting out loud using the spoken words “ten, eleven, twelve…” etc., since it could be argued that they correspond to absolute values in base infinity rather than to written numerical representations. But that’s a linguistic question. I guess it depends whether you are trying to teach people (counting one through thirty-five) or get a WTF reaction (counting 1,2,3,4,5,10,11,12…54,55).

  5. Yeah, that’s a valid concern, but I tend to think “ten” just means “the first natural number not representable by a single digit” not “the number equal to 9+1″. It doesn’t make much sense, I don’t think, to be counting in base 6 and be on 43 but to say out loud “twenty-seven”.

    Only somewhat related, but, according to what I’ve read, the words “eleven” and “twelve” aren’t vestiges of a base-12 system, but rather mean, roughly, “one-ten” and “two-ten”.

    Most people have 15 fingers? I think you mean 14!

  6. ashok says:

    Why Base 2?

  7. Binary for computers make a lot of sense because logical operations (normally) are binary choices — true/false — and computer programs are just really complicated truth trees. (That’s a really simplistic explanation, of course.)

    I’m not arguing for using binary in general, or using any base system in general; I just want to raise awareness of the arbitrariness of the base 10 system so that people will stop treating phenomena deriving from it as significant beyond the significance we attribute to them. E.g., people thinking Y2K might cause problems because of how computers had been misprogrammed in the 70s and 80s, good. People thinking Y2K might cause problems because it’s the new millenium and so the world is going to end, bad.

  8. ya mum says:

    Base 10 is such hence the increment of the next weighted column on the 10th count, so base 6 would still be called such the glyph used to describe a quantified aspect of a number system has nothing to do with the name…………

  9. What does “the 10th count” mean? What base are we meant to interpret this “1” followed by a “0” in? If you’d said tenth, I’d have known what you meant, but as is, you might mean second (i.e. base 2), or sixth (i.e. base 6), or twelfth (i.e. base C). “10” can variously mean any of those, depending on what base we’re in to begin with–and you can’t assume a default of base ten.

    So, sorry, but no.

  10. David says:

    Kant says that there is nothing intuitive as a priori that 5+7 would equal 12. In this way, there is nothing intuitive that the 1 and 0 would do the same. 9 plus 1 as a logic is the same mystery even if the symbol we use is A. Yet 10 comes from 1 stick and an esoteric 0 stick. Either you agree that you can not speak of 10 at all without what Kant calls synthetic empiricism (and where are we with that as a secondary form of truth) or 10 is merely a rest to 1 because we can not explain 111111111 + 1 as 10 without that return. So that makes that 1 stick that ticks us over a magic stick on our quest to truth through number?

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