So, A Few Years Ago, I ‘Invented’ A Number System

…or so I thought, anyway.

Like many would-be young, aspiring mathematicians, I’d ‘discovered’ a concept that was in use for decades before I’d thought of it. That concept was the factoradic number system (I’d link the Mathworld article, but there wasn’t one – I was shocked).

The number system is simple to explain and tricky to grasp. The radix of any given digit in a factoradic number is that digit’s place from the decimal point+1. (Note: There are also definitions of the system which generate digits which are always zero. I think that’s a silly approach, and furthermore, it’s not how I did it those years ago)

So, in Decimal, the radix is always 10 – each digit is worth ten times more than the one that comes before it (1, 10, 100, 1000, etc). In Binary, the radix is 2 (1, 2, 4, 8, 16, etc). The equivalent series for the factoradic numeral system is 1, 2, 6, 24, 120, 720, etc – you may recognize this as the factorials (Mathworld).

You would count from 1 to (decimal) 50 in factoradic like thus (I’m counting in 5 lines of 10 numbers each, to make the progression clearer):

1, 10, 11, 20, 21, 100, 101, 110, 111, 120,

121, 200, 201, 210, 211, 220, 221, 300, 301, 310,

311, 320, 321, 1000, 1001, 1010, 1011, 1020, 1021, 1100,

1101, 1110, 1111, 1120, 1121, 1200, 1201, 1210, 1211, 1220,

1221, 1300, 1301, 1310, 1311, 1320, 1321, 2000, 2001, 2010.

Aside from changing a person’s concept of what a number system could be, however, the factoradic system doesn’t actually do very much mathematically (at least, as far as I could ever tell from tooling around with it). It’s used to work some with permutations, but doesn’t seem to have any interesting mathematical properties aside from that.

The article’s not quite complete, though (in fact, the article’s discussion page brings this up – I guess there’s just nowhere this has been officially written out, though I’m clearly not the first to think of it).

Back when I thought I’d invented this system as a novel numbering system, I wanted it to be a full-fledged number system, so I unknowingly expanded on the work you see there in that Wikipedia article – I defined the factoradic system to account for fractions.

It functions basically the same way on the right side of the, er… factoradical point (It’s not a decimal point, it’s not decimal counting!) as on the left side. Rather than each digit representing 1!,2!,3!,4!,5!, etc, they represent 1/2!,1/3!,1/4!,1/5!, etc.

This system maintains the unambiguousness of the standard counting system, and has a couple novel attributes, as well.

A rational number is a number that can be expressed as a fraction. In Decimal and other fixed-base systems, a rational number is any number that can be expressed as a definitive series of digits or repeating digits (such as 1/3’rd, which in decimal is .3 repeating).

In factoradic, a rational number is a number that can be expressed as a definitive series of digits – all rational numbers terminate (because for any possible denominator X in a fraction, there is a factoradic digit that represents 1/X! and thus divides evenly into it). Definitive series of repeating digits are instead used to express irrational numbers – numbers which can not be expressed as any fraction (of which there are at least a countably infinite number describable in the factoradic number system).

The easiest example is the constant e (as is also conveniently noted on wikipedia in the discussion for the article), which is 10.111… repeating.

I would further conjecture that any number that ends in a definitely repeating digit series in factoradic must be an irrational number (excluding extraneous zeroes, of course).

The opposite can’t be true (that all irrational numbers can be depicted in factoradic with a definite series of repeating digits), however, due to numbers such as .00112233…, which as far as I can tell is irrational but would never repeat a series of digits. It’s a shame, since it’d be awesome if there were a number system that were capable of describing all real numbers like that.

That’s the rambling saga of my career as a would-be amateur mathematician. I’ll probably play with the factoradic system off and on for the rest of my life (as I still think of it as my own invention deep in the recesses of my mind), so maybe I’ll even figure out something novel about it one day.

Anyway, I doubt I’m the only one who’s tinkered at math, found out something they thought was astonishing, only to find that either they’d forgotten to carry a 1 or the like, or someone had beat them to the punch tens or hundreds of years ago. I wonder how common it is, and I wonder if perhaps our math education were better, or if we as a culture were more reverent of our mathematics, if all those rediscoveries could instead have been discoveries of new things instead.

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