Around here we call base one arithmetic tally marks.
What about in town?
Not so difficult as might be thought. The Romans had separate symbols for large numerals. XX with a line above each X for instance is 2000000, which is considerably more precise than our own notation.
I think you missed something on that page. XX with a line over it is 20,000. Twenty million would be |XX| with a line over it. Or perhaps better described as a three-sided box around the XX.
And it’s no more (or less) precise than our current numbering system. Maybe you meant “concise” which it is for that particular number, but is not the case for all numbers. And it does have a problem with significantly larger numbers, such as in the trillions and higher.
Why would they use base 8 and 16 for their computers as opposed to 4, 32 or 64? 64 seems like a lot but the Babylonians used a base 60.
Only if they use a place value system similar to ours, which may itself be an anthropocentric assumption.
In this post I’m using words for numbers to preclude confusion about what base they’re expressed in. Numbers in words are traditional modern human base 10=ten.
Agree that we chose base eight and/or base sixteen for computers because it was close to our human base ten.
Had we been using a Babylonian system AND used it with sixty different symbols and a powers-of-sixty place value system, then base sixty-four would have been a natural choice.
But …
A close look at the Babylonian system shows they didn’t exactly have sixty discrete symbols. Instead they had a symbol for one and a symbol for ten. Which they wrote in tight groups (*a la *tally marks)to form the larger values. If we use “o” for one and “t” for ten the Babylonian representation of thirty seven is “tttooooooo”
Switching to conventional base-ten notation …
Their quasi-positional representation for 12x60[sup]1[/sup] + 37x60[sup]0[/sup] is “too tttooooooo”.
Their quasi-positional representation for 42x60[sup]2[/sup] + 12x60[sup]1[/sup] + 37x60[sup]0[/sup] is “ttttoo too tttooooooo”.
It’s not obvious to me this really extends nicely to large numbers. And having 60 truly discrete arbitrary symbols would be a lot like using “0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwx” as a symbol set. At least for mere humans, memorizing the addition and multiplication tables for that would be tough. Quick, what is the sum of T and j? Or the product of g and 7?
Ultimately the Babylonians chose a one-ten-sixty system, not a one-sixty system. Much like the now-obsolete but still fairly recent British pence-shillings-pounds system.
Aliens with a much larger working memory set might pull off a one-sixty system and would react to our base ten about the same way we feel looking at doing addition or multiplication by hand in base two. i.e.: Waay too many digits but not nearly enough variety *of *digits.
May. But it has a lot going for it.
There’s a trade-off. If you have a system with a lot of elements in it, you can compose short sequences with a high density of information. But the downside is you have to memorize a lot of elements.
In number systems, compare base-two, base-ten, base-twenty-six, and base-sixty systems.
In base-two, you only have to elements to remember: 0 and 1. The first ten numbers are 1, 10, 11, 100, 101, 110, 111, 1000, 1001, and 1010. A million is 11,110,100,001,001,000,000 and a billion is 111,011,100,110,101,100,101,000,000,000.
In base-twenty-six, you have to remember twenty-six elements (I’ll use the alphabet as an example). So the first numbers are A, B, C, D, E, F, G, H, I, and J. A million is CEXHO and a billion is DGEHTYM.
In base-sixty, you have to remember sixty elements (0-9,A-Z,a-x). A million is 4bke and a billion is 1H9bke.
If it came to that, they could have developed a simple, thousands-based scientific notation that might look like [sub]III[/sub]DLXVII for 547 billion, then used a separator like we use a comma to mark the lower place groups (with just a blank underline for 000 sets).
Sorry, had a brainfart. I did indeed mean concise and you’re right about the erroneous notation. Oh well, can’t win’em all. 
And in ultiMayan societies, the men used a base-21 system, whilst the women used base-22.
It depends on what you mean by the “concept of zero”. Although many ancient counting systems did not explicitly have a number for zero, they used positioning or other indicators for multiples of a base raised to a power. The concept of zero is really only critical when you get into number theory and computation, and the practical applications of a concept of zero, negative, and imaginary numbers only started with physics, hence why these apparently simple concepts took so long to be discovered. Really, the critical elements in our mathematics are 0, 1, e, i, and pi, which are actually base independent, and of course all related through Euler’s identity, which is also gives us the association between exponentiation and logarithms, and from there trigonometry and continuous periodic functions.
Although the near-universal system of “counting numbers” that we use in everyday arithmetic today is decimal, there are plenty of other cultures in history that used essentially every base from 2 to 16, and some higher increments. Of course, in computation, we use base-2 as the fundamental unit of logical calculation, and octal and hexadecimal for ease of calculation and compactness in dealing with digital computers. Our adoption of a decimal base system is essentially arbitrary and doesn’t really reflect any particular scaling of natural phenomena; it is largely just a happenstance of convenience and history.
However, there is an underlying assumption here that needs to be considered; that an alien species would use any type of discrete counting of equal increments at all, rather than some kind of continuous or variable distributions of quantities. We use incremental counting systems because of how our brains are wired for numerical cognition; we see people and objects as individuals, and the way our brains work actually tries to break up continua into discrete bins for categorization. This is reflected in our mathematics, grammar, sociology, and pretty much every other aspect of human existence to the point that we struggle to understand concepts of continuous variation or probabilistic distribution, even though essentially everything we experience as the scale that we can observe it is some kind of continuum with essentially infinite variability[SUP]*[/SUP]. It is entirely reasonable that an alien form of intelligence may not make such distinctions and would develop a mathematics that does not use discrete counting, Euclidian geometry, or some of the other fundamental assumptions of our system of mathematics.
Stranger
[SUP]*[/SUP]That objects and apparent continua are composed of discrete atoms and molecules, and below that of fundamental particles, is not evident to the point that our description of mechanics has only reflected this in the past couple of centuries, and it was just over a century since we’ve experimentally verified the atomic nature of matter. Most of our practical evaluations of structural and fluid materials still assume that they are a continuum, often using rules and properties that are empirically determined due to the essential difficulty in calculating and observing behavior at the level of individual molecules or atoms.
I remember reading a nice puzzle in Scientific American one time. It was a version of the ‘Try and work out what numbers these symbols represent, and what base they are using’ from a mathematical equation some astronauts found scrawled on a wall on a distant planet. The only clue was that they knew the aliens had manual dexerity and formed their base-number system from the digits on their ‘hands’.
When the answer was given - “Base-9” - there were protests from many readers asking "How can it be Base-9? How many digits do they have on each arm?’
The answer of course being 3.
Did the Romans need number notation to describe numbers past a few millions? Their equivalent of an accountant, quantity surveyor or whatever isn’t going to have needed notation for billions, surely?
The number of digits you have need not be the same as the number of distinct symbols. The Mayan system, for instance, had three symbols. Their zero was zero (it looked sort of like a seashell or maybe an Easter egg), but all of their other digits were composed of combinations of horizontal lines and dots. One dot was “1”, two dots was “2”, and so on, until you got to a line was “5”. Then you’d have a line and a dot for “6”, and so on, until three lines and four dots formed the digit for what we’d call “19”.
And the number of digits that’s convenient depends not just on physiology, but on psychology. Maybe, for instance, the aliens are better at distinguishing slightly different symbols than we are, and better at learning a large number of symbols, in which case they might not think that a hundred different symbols was a big deal. Or maybe even a thousand: Even among humans, speakers of Asian languages learn more symbols than that (though of course those are mostly more complicated symbols than the Arabic digits).
Its not a zero. Saying that just confuses the matter. It’s the idea of a placeholder number, and you do have to have a zero number for that, of course.
So if V=5
In a placeholder system
VV=55
but otherwise
VV=10 (aka X)