We count in tens, so we are told, because we have ten fingers.
A creature from another planet, with a different number of digits, would count numbers in a different way.
When trying to explain Number Bases to a class of eleven year-olds I came up against one major problem: what do we call "10"? For whichever number-base we might want to use, we need not just number symbols, but also number-words.
Not knowing any aliens, let's take a dog, (one capable of counting) as an example.
I suggested to the class that we imagine ourselves on Sirius, a planet ruled by intelligent canines. We count on the five digits of our hand, maybe the dogs would count on their claws?
A dog has four paws, so the children decided he would count in fours, not tens, and use "four" as his base.
Just as we write "1, 2, 3, 4, 5, 6, 7, 8, 9" and then put "10" to mean "one group of ten and no units", so our dog will think "1, 2, 3" and then put "10" for "one group of four (one complete dog) and no units".
We can't say his "10" is "ten", because it stands for four. We need number names as well as digits, so let us call this "10" "doggie".
Now the numbers run:
one, two, three, doggie (in words)
1, 2, 3, 10, (in symbols)
doggie-one, doggie-two, doggie-three, twoggie (ie two-doggie)
11, 12, 13, 20,
twoggie-one, twoggie-two, twoggie-three, throggie (three-doggie)
21, 22, 23, 30,
throggie-one, throggie-two, throggie-three.... Now what?
31, 32, 33 ...
If you've followed the argument so far then you'll realise that since 1 + 3= 10 (one plus three equals doggie) in the dogs' base four system, then 1 + 33 = 100, and all we need invent now are names for 100, 1000 and so on (should we need them). In any base "10 x 10 = 100"; but if the base isn't ten we can't call "10" "ten" or "100" "one hundred". The children thought a suitable name for 100, in the dogs' system would be "houndred". So now we can write base four numbers in symbols, and in words - for example 231 is two houndred and throggie-one.
Children enjoy being amused in this fashion, and it all helps put across the idea of Number Bases. We also did battle with a clever octopus, and base Eight (with suitably silly names such as "soxty" for 60 base eight) and went on to extend the idea to bases greater than ten. It didn't take much to demonstrate that if bases less than ten needed less symbols but more columns to write a given number, then base twelve and other bases greater than ten would need more symbols and less columns.
In the dog's arithmetic we count to four before we write 0 for the units and carry a unit (a doggie) to the next column to the left.
1 + 3 = 10;
23 + 2 = 31 (You can check this, if you want, by remembering that twoggie-three stands for two fours plus three units - i.e. eleven in base ten; and throggie-one stands for three fours plus a unit, or thirteen in base ten).
By the time we finished with base four, and the dogs, one or two of the children had even produced illustrations of doggie postage stamps numbered in base four and with values in doggie currency, ( 1D = 10p, or 1 Doglar = 10 paws; the doglar being a doggie dollar... And there were some awful jokes about dividing the Sirian year into dog-days, the day into dog-watches and the police into watch-dogs!
Anything you can do in base ten you can do in base four: add, multiply, subtract, divide - though you'll have to do the work yourself since calculators are geared to decimal and won't do complicated sums in other bases.
Here's the dog's multiplication table:
- read the answer where row and column cross; for example 2x3 = 12.
Armed with this table, you can do any multiplications you like - long or short - but don't forget you're working in fours and not tens!
Here's a "long multiplication", for example - 123 x 12
In base ten we use 9 symbols: 1, 2, 3, 4, 5, 6, 7, 8 and 9, and the zero: 0, the symbols telling us how many units, tens, hundreds etc we are counting, while the position of the symbol within the given number tells us whether the items we are counting are tens or hundreds and so on.
This is called Positional Notation and though we use it to count in tens, ten being our counting-group or base, we can in fact use any number we like as our base.
If the base is ten then we need nine symbols, each representing a number less than ten, and the zero. We need the zero to show when there are no tens or hundreds or whatever. If instead we want to use base four, then we need the symbols 1, 2, 3 and the zero, and if base twelve we need two extra symbols (one to represent "ten" and one "eleven"). The symbol "10" can stand for any number you like. Most people know something about Binary (base two), Octal (base eight) and Hexadecimal (base sixteen) because the computer has made them important.
The fact that "10" can be any number has prompted some to ask if there is an optimum base for humans, and then to wonder if that base should replace base ten. There are many who believe that base twelve is better than base ten, and you'll find reasons in the articles on this site.
What about "decimals" though? To divide by ten in base ten, you can just move the decimal point. So in base four, to divide by four ("doggie") you can move the doggimal point. 12 /10 = 1·2, for example - twelve divided by ten is one point two (one and two tenths); or, if this is base four, doggie-two divided by doggie is one point two (one and two doggieths [or fourths]).
Just for comparison:
one half = 5 tenths = 0·5 (base ten) = 2 fourths = 0·2 (base four)
one quarter (or fourth) = 25 hundredths = 0·25 (base ten) = 1 fourth = 0·1 (base four).
Lastly, if 10 means four, then 100 = 10 x 10 = four x four = 16 (base ten)
1000 = 10 x 10 x 10, or 103(base 4) = 43 = 64 (base ten), and so on.
Glaser's book on non-decimal numeration quotes Weigel's suggestions for base four
10 = Erff, 20 = Zwerff, 30 = Dreff, 100 = Secht and 1000 = Schock (this last being German for "sixty items" serves here for sixty-four).