The Permutational Numbering System
The first mathematical concept we all learn is counting: one, two, three... ten, eleven, twelve... twenty, twenty-one, twenty-two... one hundred, one hundred and one...
Later, when we started meddling with computers, we all discovered that counting wasn't the simple thing we always thought. Computers don't count 1, 2, 3. They count 1, 10, 11. They use a different numbering system: binary, or base two. And you might have needed also hexadecimal, or base 16. Our dear old way of counting turned out to be decimal, or base ten. On all these numbering systems, there's a "base", a special number we need to know in advance to understand which number are we talking about. Let me put an example: 1011. Now, is it necessary to use always a base? The answer is: no. I've discovered a new numbering system (and I swear I've discovered it independently of anybody else who might have discovered it first) where no number is given a special status. I call it permutational, and counting is like this:
1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 1000...
Do you understand how it works? The last digit can be either 0 or 1, the second last can be 0, 1 or 2, the third last can be 0, 1, 2 or 3... etc. And what does 1011 mean? 1011=1*4!+0*3!+1*2!+1*1!=twenty-four+zero+two+one=twenty-seven
(4! isn't a good-looking four, it's 4*3*2*1)
Perhaps you don't quite see why this works like this. You can believe me with blind faith, or you can try to work it out by yourself. Anyway, I won't bother proving any of the statements I will make about the permutational system: it's boring, and I'm already pretty sure they're true. But if you have any comments about them, I'd love to know.

What's the use of a new numbering system?
The first advantage is very very obvious. Factorials are expressed very simply: 3!=100, 6!=100000.
Another interesting fact, only slightly less evident, is that there are rules of divisibility for all numbers. To know if a number is a multiple of, let's say, seven, you only have to look if the last six digits make a multiple of seven. This can be very useful with huge numbers and small divisors.
Now, let's take a look into the realm of permutational numbers.
This is a list of the squares (n^2):
1, 20, 111, 220, 1001, 1200, 2001, 2220, 3111, 4020, 10001, 11000, 12001, 13020, 14111, 20220, 22001, 23200, 30001, 31220, 33111, 40020, 42001, 44000
It doesn't tell you anything? OK, let's take a look at the last digit: 1, 0, 1, 0, 1, 0...
And now, the second last digit: 0, 2, 1, 2, 0, 0, 0, 2, 1, 2, 0, 0...
And the third last digit:
0, 0, 1, 2, 0, 2, 0, 2, 1, 0, 0, 0,
0, 0, 1, 2, 0, 2, 0, 2, 1, 0, 0, 0...
What happens with the cubes? Let's see:
1, 110, 1011, 2220, 10021, 14000, 24101, 41110, 10111, 121220, 150121, 222000, 301201, 344110, 440211, 540220, 644221, 1103000, 1230301, 1403110, 1550311, 2043220, 2251321, 2511000
Last digit: 1, 0, 1, 0, 1, 0...
Second last digit: 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0...
And the third last digit? That's harder to see. The period is 24, instead of 12 like in the squares. Can you guess why?
Let's have a look at another interesting collection of numbers: the powers of two (2^n):
1, 10, 20, 110, 220, 1110, 2220, 10110, 20220, 41110, 122220, 250110, 540220, 1421110, 3142220, 6330110, 15000220, 32001110, 64002220, 140010110, 280020220, 570041110, 1150122220, 2310250110, 4620540220
And these are the first powers of three (3^n):
1, 11, 111, 1011, 3111, 20011, 100111, 301011, 1203111, 3620011, 13500111, 43101011, 141303111, 434220011, 1315000111, 3947001011, 10855003111, 32577020011, 97755100111, 251277301011
Probably you have noticed already that bolds, italics and underlinings aren't pure decoration. Curiouser and curiouser, isn't it?

Numbers smaller than one
Of course, a numbering system wouldn't be complete if it couldn't express fractional numbers with the help of a point. There is no problem for this: 0.1011=1/2!+0/3!+1/4!+1/5!=11/20
The same as before, the first digit after the point can only be 0 or 1, the second 0, 1 or 2, etc.
In permutational notation, it's fantastically simple to know if a number is rational or irrational. Just count how many digits are there after the point. If there are infinite, it's irrational; if there are finite, it's rational. Some irrational numbers are particularly beautiful in this numbering system:
e=10.111111111...
1/e=0.02040608...
cosine of 1 radian=0.10045008900...
sine of 1 radian=0.120056009...
If n is a prime number, 1/n has n-1 digits after the point, and the last one is n-1. For example, 1/5=0.0104

If you're still interested...
There are still lots of unsolved problems in number theory. For example: take a number. If it's even, divide it by two. If it's odd, multiply it by three and add one. Take the result, and repeat the same operation. If you do this enough times, do you always end with a one? Computers have always ended with a one with every number they've tried, and many eminent mathematicians have tried to prove that it must be always so, but without success. Maybe it was just a matter of using the wrong notation.
If you start playing with permutational (and be careful, you can get hooked!), you'll soon find there's only one drawback: you would need an infinite supply of different digits to be sure you can express any number. Usually, this shouldn't be a problem. You're not very likely to use numbers with more than ten or twelve digits (12! is approximately equal to five hundred millions), and with the usual numbers from 0 to 9 and a couple of letters, A and B, you would have enough. If you really want to work with huge numbers, there are ways to shorten them to a comfortable size: the same way that in decimal we've got a scientific notation, and five hundred millions becomes 5*10^8, we can use factorials in permutational, so five hundred millions is approximately 1*12! (or 1*200! for the purists)
Still, there could be some trouble if you want to calculate the first forty digits after the point of pi. You could need all the digits from 0 to 9 and all the letters from A to Z, and still not have enough. But with a bit of imagination you can also overcome these problems (Hint: have you noticed there are parentheses in your keyboard?)
If you get any interesting result experimenting with permutational, or if you don't, but would like to talk about the interesting things you thought in your failed attempts, e-mail me: lusina@redestb.es

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