…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.

## I Despise Unrealistic, Nostalgic Romanticisation of the Past

October 22, 2009So, since this guy was smart enough to turn off comments from people who might disagree with his wistful but not particularly factual diatribe about American history, I’m going to make this post a comment in response to it.

I see one person waxing nostalgic about their youth, creatively equating their actions with those of a select few a couple hundred years ago, and acting like this form of behavior is one that was somehow thriving and is now dying.

This article reads like one of a million articles discussing the Moral Decay Of The Youth During This Generation – and it’s not even good with the details.

Americans weren’t fighting for individual freedom – they were fighting for the ability to represent themselves in government. That’s what “Taxation without representation” was about, our original demand of England was for us to receive representation in Parliament as citizens of England, so we could have a say in our government. Not “My” government – “Our” government. Not individual freedom, but our self-determination as a people. Yes, there’s a difference. Yes, it’s important.

Nor was our government forged from ‘understanding of human nature’ any more than it is today (and what understanding we had then, wasn’t as good as what we have now anyway). Our government was created, in fact, from countless minor compromises and ideological conflicts of every type, some with unspoken underlying issues associated with them, and all of this parallel to some individuals seeking their own personal profit and using the events of the times as a vehicle to obtain it. Meanwhile, the people got preached at by the aristocracy controlling the flow of information, mostly following their lead, and being smacked down when they deviated – the proportion may be different, but the parts of our government are all the same today.

Furthermore, ‘our founders’ were not remotely some holistic monolith of thought and beliefs fundamentally distinct from what exists today. In fact, they almost immediately polarized into two camps, one of which advocated strong centralized government (the Federalists, who were vaguely similar to liberals today), and the other of which advocated weaker decentralized government (the Democratic-Republicans, which we could compare to conservatives today). Some didn’t like slavery, and others thought the Bible advocated it.

Finally, the founders were not all religious, just like they didn’t all believe the same ‘holistic’ junk. Thomas Jefferson even wrote an entire gospel which explicitly excluded all miracles, bucking the fundaments of even the modern Christian faith and effectively creating a philosophical, rather than theistic, Christian text.

Our nation’s founders were people, just like us. They bickered and argued and got into physical fights with each other, and some of them sure as hell did profiteer from the War of Independence (most notably, through smuggling). One wanted the Turkey to be our national bird. One was so incredibly popular, he was

literallyasked to be our King (Washington, if you’re curious).When people romanticize these people and try to wax philosophical about how much better “things were back then”, they’re ignoring the reality of those times, and more importantly, they’re ignoring the reality of the present. America is little different in its’ fundament than it was two hundred years ago, with one big exception: Businesses are more powerful than the government is now, so they run the government. There are two ways to fix it, and one involves making the government really powerful, and the other involves destroying some businesses.

Most importantly, even back during the Enlightenment, ignorant blowhards were complaining about how social progress was Destroying All That Was Good. They, most of all, have not changed.

Tags: Commentary, Founding Fathers, Moral Decay Of The Youth, People Who Do Not Know What They Are Talking About Trying To Sell Books To Make Easy Money, Peter Breggin, Rant

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