I've often wondered about the possibility of using something in the spirit of a tensor to deal with the pros and cons of various number bases, although I certainly don't have the expertise in either tensors or number theory to flesh out a proper system. For those unfamiliar (and take it with a grain of salt, I don't really know what I'm talking about) a running theme in tensor calculus is the refusal to pick a coordinate system and instead focus on the quantities and objects which remain the same when transforming between two arbitrary coordinate systems.
Am I the only one who thinks it's a bit nuts to seriously consider ease of using fingers as a basis for a numbering system?
It may sound a bit nuts at first, but its actually a far more central aspect to the whole discussion than you may expect. The arabic numeral system is one of the more profound mathematical leaps forward in history. As an amusing exercise, try brushing up on roman numerals and then attempting to write some medium to large numbers. You'll find there is no general correlation to the length of a number string and the size of the number. Its even more unwieldy if one attempts to do arithmetic on these objects. But still the question of "why ten?" While it is certainly possible to formulate a similar system in a different base consider two further points; the ease with which we can teach arithmetic to children, and the linguistic considerations of a culture. About the first point; every child comes equipped with a built in abacus in base ten. Its this biological fact that allows for us to teach basic arithmetic at such a young age. Early exposure is the first step in getting these things to become automatic and finger counting for a very young child is a highly important tool. To be fair, the linked article has an alternative suggestion to this which uses sections of fingers instead of fingers to count. I would argue that in the mind of a child it is a far easier and less subtle idea to count with whole fingers instead of sections. I'd also posit that the younger a child the harder it is to explain a concept like sections of fingers. Whole finger counting is so intuitive that I'd bet there are kids out there that didn't even need to have it shown to them. As for the second point; there is a really funny correlation between the cultural development of our language and the way we think about numbers. I read a book a few years ago on the history of mathematics and in it the author discusses the findings of some historical anthropologists. They studied various indigenous peoples who had remained relatively isolated from the rest of the world. Their findings showed a surprising correlation between the warmth of the climate and the sizes of numbers they had words for. Cultures in very cold places would often only have words for 'one' 'two' and 'many' while people in warmer climates could often go as high as twenty. Their conclusion was that there was a direct correlation on the number of digits available to count with and the sophistication of a peoples counting system. If you wear gloves and boots all the time its kinda hard to finger count. Conversely, if you have sandals on, there's whole 'nother row of ten piggies to utilize.
Nah. We're too entrenched into base 10 at this point. To change now would be ludicrous. I mean, just look at America when we switched over to the metric system. What's that you say? "America doesn't exclusively use the metric system?" I rest my case.
> Moreover, with base-12, we can use these three most common fractions without having to employ fractional notations. The numbers 6, 4, and 3 are all whole numbers. On the other hand, with base-10, we have to deal with unwieldy decimals, ½ = 0.5, ¼ = 0.25, and worst of all, the highly problematic ⅓ = 0.333333333333333333333. WTF?
Wouldn't the dozenal system have their own fractions with this same kind of problem? Overall, in the grand scheme of things, base 10 and base 12, whichever you use, gets you to the same answers. That said, this guy right here explains the dozenal system so charismatically. Even though the video is almost ten minutes (HA! BASE TEN! EAT THAT!), it goes by so quickly.
I think I was a little unclear above. Let me try this again: The article tries to make a statement about the superiority of base-12 by pointing out that you can divide 12 by 2, 3 and 4 and get integers as result. Then it tries to show how bad base-10 is by dividing ONE (not ten) by 2, 3 and 4. And while that would be a decent argument had the author used ten (only 10 / 2 = 5, the others are still rationals), I think it's WTF worthy to use one instead here.
I'm unconvinced that a base-12 numbering system is particularly better than base-10, but I can explain what the author was trying to say about fractions and decimals. In base-12, one half would be represented as 0.6, one third as 0.4, and one quarter as 0.3. In base-10, the number immediately following the decimal mark is the tenths place, so one half is 0.5, or five tenths; in base-12, that first digit after the decimal mark would be the twelfths place, so one half would be 0.6, for six twelfths.
This is an interesting topic and in the article they cover a lot of important points like the ability to easily divide and count on fingers. This all seems great but i still think it's easier to count using one finger for one number and it's also easier if you are trying to show someone at a distance. You can easily see somebody holding four fingers in the air but it's difficult to see which finger and which part they are pointing at. The article also points out that we already use dozens for plenty of things but i can't say that i see a reason why not to use both systems at once. Another counting system i find interesting is finger binary although it is fairly impractical.