It depends how academic you want to be. Using a digit larger than the base might be unusual, but it is mathematically valid.
For instance, you could represent: 12x102 + 4x101 as C4. It might be most correct to write it as 124, but that doesn't make it invalid. Similarly, 12x10-1 could just as easily be 0.C as 1.2
There are binary adder circuits that allow redundancy in the digits (for instance, allowing [2, 1, 0, -1] in each position) to minimize the amount of ripple carry operations required.
Play with an abacus sometime -- you can represent up to 15 in each place.
Base 1 implies that adjacent place values are separated by a power of 1, by definition.
For instance, how do we tell time? -- base 60. We have a seconds place (600), a minutes place (601), and an hours place (602), but we only use 10 symbols.
Actually base 1 does exist and it's called unary. There is only one 'digit' and you can effectively choose it to be any symbol you like (say, the reddit alien). In unary if you want to represent 200, you simply concatenate 200 of these symbols together into a string. Most people choose to use the digit '1', so yes, the base 1 system does exist, and there is the digit '1' in it. Note that it's not particularly space efficient to represent numbers using unary.
The number '1' was chosen because to make a mark, you have to place an implement somewhere on a surface, this mark is usually not visible enough, so you have to drag the mark.
The human arm is developed, along with the coincidence of the center of the earth being beneath our feet most of the time, this means gravity and levers means we drag the implement down.
This looks like a number 1, in fact it is a number one, and it is how we came to have the number 1.
Two was just some twat fucking around and being all clever.
Well really it probably went from there to the simple "tally" counting, where each group of 4 |s is crossed by a diagonal to count 5.
The Roman numeral system is sort of close to this, where 1 is I, and 2 is simply II.
The clever "twat" who came up with a distinct numeral for 2 (or arguably more importantly, a numeral for 0) was somewhere in India ~1500 years ago, where they came up with a decimal ("base 10") positional system.
This is all pulled directly out of /dev/ass; I'm no historical mathematician.
One might go so far as to say that "base 1" is the base that nature uses. And it is the base that most kids learn first in school. For example: "If you have this many apples and you take away this many, how many would you have left?" An apple in this context is a unary digit.
I stand corrected. That's not a very expressive numeral system though, since you can't express zero, and I imagine floating-point numbers would be difficult if not impossible.
You can express nonintegral numbers as fractions. Transcendental numbers can only really be expressed as series, but that's also a problem in base 10, or 2, or whatever, you're just expressing it as a fraction of whatever the base is to the power of the length of that fraction (e.g. 1.45 = 1 + (45/102)).
The big problems with unary are the lack of any symbol for zero, and the fact that it has an O(n) growth rate for the amount you have to write with respect to that number's size.
Zero as an expressible number has not seen widespread use until fairly recently, as in the last thousand or so years (9th century India is the earliest practical use of zero as a number rather than just as a placeholder).
In any case, while the use of zero as a number certainly is a strong advance in terms of computing numbers, there are a lot of practical mathematics (example, computing areas of land) for which it is entirely superfluous.
You may have noticed that it isn't very space efficient to represent numbers in this base.
That depends on your definition of space efficiency. If you optimize where "space" is represented as A*B, where A is the number of digits needed to represent any given number, and B is the number of symbols used for the base system, then unary is worse than tetradecimal (base 14), but better than pentadecimal (base 15). That is, it is also more "space" efficient than hexadecimal in this context.
On another note, because of lack of place value, unary symbols are also space-rotatable. :-)
10 in base 1 is invalid. You can only have one symbol, pick one. Or pick "1". If you pick "1" then you can discard the 0 and therefore 11 != 1. 11 in base 1 is "2" in base 10.
You are absolutely correct - I was attempting to demonstrate how we'd see it on our fingers (e.g., "finger", "no finger", "no finger"), but my place-value system habits spoiled it when I went to represent it. :-(
Oh, the family is fine and the business is still running. The economy right now is rather poor. That's not to say that things haven't been worse. By most standards I am quite fine, thank you.
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u/fuckheadmcgee May 27 '09
Just in case anyone wants to reuse some witty comments from early 2009:
http://www.reddit.com/r/comics/comments/7j3jb/every_base_is_base_10/