A Basic Question[about math]--edited title

Factual Questions
1

Forgetting cultural reasons as to why we use Base 10, I am curious. If we could start over, from a mathematical perspective, what base would make the most intuitive sense from a scientific point of view to use?

Base 2?
Base 10?
Base 16?
Base 1,A45,Q3$?
Base something else?

Relating to this question, have there ever been any mathematical (or physics I suppose) questions resolved by converting from Base 10 to another base where answers that were elusive before became more readily apparent?

Does base matter? Do all your base belong to us? (sorry)

2

Base 10, as in fingers and toes

3

base 10 allows for everyday number use which doesn’t have either many places or a large character set.

4

Well, as far as computers are concerned hex is the most commonly used only because conventional computer hardware operates uses binary switching. Binary is unwieldy because the numbers can get long, but hex is nice because 8 digits covers a 32-bit word.

But this advantage doesn’t apply to anything else AFAIK. Math and physics are concerned with models, equations, what have you, but other than physical constants, theories don’t much care about what base the numbers are in. Except for dimensionless constants, like Avagadro’s number, the units are arbitrary anyway so changing units changes the number (like expressing the speed of light in miles per second vs. kilometers per second).

The higher the base, the fewer the digits required to express a number, but the number of digits doesn’t seem to bother anybody.

5

Base 16 worked well for the Babylonians as well because 16 can be evenly divided into halves, quarters and eighths. 10 doesn’t divide evenly by 4 or 8. 12 is also a ‘better’ base for the same reasons, but isn’t used except for the collections we call ‘dozen’ and ‘gross’
Also the base 60 system currently used for time has held up pretty well over the years even though those pesky French thought otherwise for a few years.

6

One advantage to using a different base is it reduces the number of fractions required for common measures (I believe this is one reason why Babylonian astronomers developed a base 60 system for computation). Base 12, for example, allows for non-fraction division by 2, 3, 4, and 6.

7

Incidentally, as I understand it, Avogadro’s number is an arbitrary feature of human unit selection as well (the number of atoms in 12 grams of carbon-12). There is no reason that another culture would find this same number to be of any importance.

(This isn’t to disagree with your main point.)

8

Sure, but base 10 is essentially the same as base 1000, say, which is quite a high base requiring very few digits indeed… (on the flip side, those digits are each three times larger in size, making it a wash, and illustrating why no one cares about the illusory difference in digit-lengths between bases).

Sure, but base 10 is essentially the same as base 1000, say, which divides evenly by both 4 and 8.

The practical advantages of base 12 and base 60 over base 10 aren’t in divisibility by 4 and 8, as such, but in divisibility by 3.

9

Heh.
You are right.

10

Right.

For that very reason, I’d prefer base 6 over base 12 (if that were the choice), or base 30 over base 60 (ditto). Those extra factors of 2 are just complicating things.

Like making the multiplication tables four times larger.

11

My vote is for base 8.

Mainly I say this as a computer engineer who would prefer that things like binary-coded decimal never existed. (And don’t even get me started about the representation of IEEE754 floating-point numbers.) A base that is a power of 2 is very convenient to use when working with machines whose optimal mode of operation is binary. Eight is nice because it has a number of digits that is small enough to be easy to remember and distinguish visually, but that is large enough that useful numbers can be expressed without requiring many places.

The next lower power of two is four, which is too small. The next higher power is sixteen, which might also work; but IMHO that’s just a little too large.

12

I think you’re onto something. An ideal base would cover a number of small primes, but none more than once, so you can get as many prime factors included without exploding your number of different digits needed. Let’s say that you also want more than one prime factor in your base–base 17 would really suck. So your main candidates are:

6
10
15
21
30
42

We’re probably already drifing into goofy with base 42, so I’ll stop there. Assuming that you want 2 as a factor as well (halving is such a common mathematical operation!), that leaves:

6
10
30
42

Base 30 and 42 would be pretty unwieldy (in base 30, you have 900 cells to memorize in your multiplication table); 6 and 10 both look pretty good from a practicality standpoint. I think base 6 is probably a bit better–it has the two smallest primes, rather than the smallest and the third smallest–but it seems to me that the base we ‘randomly’ settled in based on our number if fingers is pretty close to optimal.

13

For everything except computer engineering (see below), I think base 12 would be awesome. In base 12, 1/2 is .6, 1/3 is .4, 1/4 is .3, 1/6 is .2 – all nice, even decimals (err, duodecimals?).

They are not three times larger, but 100 times larger. Can you imagine having to memorize 1,000 unique symbols to be able to read and write numbers?

As an ex-computer engineer, I’ve always hated octal. With the single exception of the old CDC/Cray systems and their 60-bit words, I’ve never encountered a system where the bit-length of their words was divisible by three. Hex comes out even on a 16-, 32-, or 64-bit system, even if you start at a byte boundary that isn’t an even word boundary. Octal doesn’t.

14

They are three times larger in terms of information-theoretic size: that is the whole point of the observation that base 10 and base 1000 are essentially equivalent. Whether you call “784” three decimal digits or a single base-1000 digit is purely a matter of taste, making no difference to the mathematical reality. (Of course, this takes advantage of the fact that each of the 1000 digits in base-1000 admits a further decomposition into three decimal digits, rather than giving each its own arbitrary symbol, but sure, why not? It’s still considerable as base-1000, in the same way as hexadecimal digits admit decomposition into four bits apiece while still being considerable as base-16).

15

I had 20 of them, last time I counted!
What would a base 20 numerical system look like?

16

Like base n for every other n; it’s not particularly special. The Mayans used it, though (observe, in keeping with my previous point, that they furthermore decompose each base-20 digit into a pair of a base-4 and a base-5 digit, these in turn each being given in unary notation).

17

I think the reason we’ve accidentally fallen on 10, aside from the fact that it’s the number of fingers we have, is the size. Any base we choose is going to be a balancing problem between the set of symbols and the length, in symbols, of the representation of common numbers. The obvious examples would be that base 2 is clearly too small because even a common number like 20 requires 5 digits, where a base like 60 requires memorizing and recalling that many different symbols. Now, I don’t think the number of symbols is strictly limiting because most languages have alphabets at least 2-3 times larger than the number symbol set. As others mentioned, the real problem is the limiting factor of being able to memorize and quickly recall simple arithmetic which gets harder at the square of size of the symbol set. So in that sense, I think 10 is a pretty good compromise on the set of symbols. So I’d say chances are any base chosen ought to be relatively close to 10.

The part where I think base ten fails is, as others mentioned, is that it’s not divisible by a lot of common fractions. Unlike SCSimmons, I don’t think limiting to just primes is a necessary constraint because even though 4 isn’t a prime, it’s probably more useful than 5 for most cases, and certainly more useful than 7. I saw a method somewhere where the author was arguing for a base 12 that related the size of the base to the number of divisors, so, for instance, base 10 gets a ratio 5, where base 8 and base 12 are both relatively close in size to base 10 but have better ratios (4 and 3 respectively), and are divisble by more useful values (4 alone is more useful than 5, 12 has the added bonus of being divisble by 3 as well). The next closest numbers to 10 that have a better ratio are 6 and 24, both of which probably fail the size problem. So based on this, I would argue that base 8 and base 12 are probably the two best candidates for bases.

Finally, with the prevalence of computers, being a direct power of 2 is probably somewhat of an advantage, but probably a very small one since dealing with base 10 isn’t a problem with modern computers. I don’t think it’s enough to overtake the better ratio and thirdability (is that a word?) of 12, but it’s still a slight nod in it’s favor.

So if I were to choose an optimal base, I’d go with base 12 followed by base 8 not too far behind. I’d say base 10 is probably third. Finally, I think base 16 and base 6 are roughly tied for 4th. Base 16 suffers from being a little large and having a slightly worse ratio than 10; however, it does have the benefits of having a better set of divisors (4 is strictly better than 5, IMO) and being a power of 2. Base 6 has a good ratio and is thirdable, but it’s almost certainly too small and strictly inferior to at least 12 (same ratio, divisible by 4, better size).

So I’d say we didn’t do too bad with choosing 10, but probably could have done a little better.

18

I agree; you want a base which is divisible by various primes while also being as small as possible (thus, it might as well be squarefree). I never thought about it before, but your observation that this makes base 10 actually quite good is an interesting one.

19

The point is that, base-10 already handles division by 4 quite well; just re-think of it as base-100, and there’s no problem. That’s why the restriction to primes enters; to make the base as small as possible while still covering as many factors nicely in this way as possible, you might as well not bother worrying about covering composites directly; as long as you cover each of their prime factors, a suitable power will cover the composite itself.

20

That’s a good point, though if we did use octal regularly, we probably would have word lengths frequently divisible by three. A nine-bit byte, perhaps. :wink:

Although there is probably something to be said for a power-of-two base where the power itself is also a power (or at least a multiple) of two, which I agree makes hexadecimal very attractive. I know that I would prefer to work mentally in base 8, however.