This has been a thought in my head since round about 1979; I've never sat down to write it, though I have remembered it from time to time. I remembered it yesterday - and though it isn't new, as I just today found the same observation elsewhere on the net, I'm going to go ahead and write this anyway. Call it fan service. The post below is based loosely on an essay by Isaac Asimov (but the conclusions are mine).
A 'number system' is a means of expressing calculations based upon the number of symbols used, in order to produce natural numbers - numbers used for counting. We use what's call the 'decimal' number system, or base 10, in which ten symbols are employed: 0, 1, 2, 3 and so on. The binary number system only uses two symbols, 0 and 1.
So let's take a number, say "54." We don't normally think of it this way but the structure of the number represents two columns: the "5" is in the first column and the "4" is in the second column. Express in scientific notation, the first column equals 5 x 101 while the second column equals 4 x 10 to the zero power (100). Anything to the zero power is equal to 1. Feel free to read further on exponents on Wikipedia if you so desire.
Let's say you want to multiply 18 by 3. If you weren't doing this though the use of a tool, you would start by multiply 3 by 8. You would then divide the product by by 10, putting down a "2" in the 101 column and a "4" in the 100 column. Then you would multiply 3 by 1 - and since the "1" is already in the 101 column you would again divide by 10 and add the product, "3", to that column:
The principle is the same with a 2-based number system, only then the columns are based on 20, 21, 22, 23, 24, 25 and so on. In ten-based numbers this translates to 1, 2, 4, 8, 16, 32 and so on. Written in 2-based numbers, however, this is 1, 10, 100, 1000, 10000 and 100000. Each column outwards from the right represents an exponent of 2.
If we want to translate "54" into a 2-based system we would begin by dividing 54 by 32 (1 with a remainder of 22), then dividing 22 by 16 (1 with a remainder of 6), then dividing 6 by 4 (1 with a remainder of 2) and finally dividing 2 by 2 (with no remainder). We don't need the "8" column (23) and we don't need the "1" column (20). Let me express the number using the same sort of table as above:
This is old hat for a lot of you - but believe it or not, some people have never stumbled across this before. Others took this in school but never really got it, you know? Some people aren't getting it now, but - well, we do our best.
So 54 expressed in a binary number system equals "110110."
Very well. Let's try multiplying six times nine in a 13-based number system. We typically think of 6 x 9 as "54;" but in a 13 based number system, the columns are 131 and 130, or "13" and "1."
Therefore, we start by dividing 54 by 13, not 10; this gives us 4 with a remainder of 2. The 2 then fits into the 130 column, so our product is 42. 9 x 6 = 42.
This was pointed out to Douglas Adams, who answered, "I may be a sorry case, but I don't write jokes in base 13." Pity. This is the sort of thing that comes back to bite writers in the ass. We sit at our desk, stare into the garden and concoct a number - and then it turns out somewhere down the line that the number means something.
Personally, I choose to disregard Adams' explanation. Obviously, the number did not come into his head at random. It was placed there by a 13-fingered God who was trying to send a message to the modern world. The universe may not fundamentally make any sense, as Arthur Dent noted, but it isn't because 9 x 6 doesn't add up to 42. It is because God has more fingers on one hand than he has on the other. God is a freak. God is an outcast among his own people, creating this world in order to assuage his deep-seated need for both affection and total obedience. He is a shallow, hurting, abused boy stuck in his room, afraid to go out where he will be taunted and ridiculed, so instead he hides and haphazardly mis-fashions a world (his left hand, the Left Hand of God, has the extra finger and winds up fucking up everything the right hand tries to build) while we remain blissfully unaware of this all-too-important defect.
And millions of Adams' fans missed it. Perhaps the greatest theological proof in the universe and people missed it!
God must be so pissed.