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Sunday, December 14, 2008

A Century of Posts

This is my one hundredth published post, so I thought I would commemorate the occasion by talking about why one hundred is such a special number. The specialness of 100 is primarily rooted in our 10-based numerical system. Basically, 100 is nice because it is a large, uncomplicated number but not unmanageably large. It is also 10 x 10, making it a perfect square. However, we are so used to thinking in base-10 that we forget how much it colours our thinking of numbers. I suppose, to be entirely responsible, I should start with a description of what a base-n system of numbers is.

Any number which we can write as a decimal can be written as an additive sequence. I briefly covered this in my post on the Cantor Set, but I will repeat myself here to refresh your memories (also, please note that I will now use * for multiplication rather than x, simply because it is what I more used to writing on the computer). For example, the number 15.34 = 1*101 + 5*100 + 3*10-1 + 4*10-2. The base of your system is whatever number is being raised to the exponent in the expanded representation of the number. There are some other base systems that receive widespread use, the most famous and popular being the binary system (base-2). When looking at the Cantor Set, we used the ternary system (base-3), and in many computer science problems it is useful to use base-8 or base-16 (base-16 gets somewhat awkward to use since we need additional symbols beyond 0-9 to represent values up to 15. The letters A-F substitute in an unpleasant mixture of letters and numbers, which is why, despite my fondness for the number 16, I resent base-16 (also called hexadecimal)). Those, of course, are not the only possible systems, since one could potentially choose any base.

If we did not use base-10, then, 100 would cease to be such an exciting number. While it would still have the property of being a perfect square, it would no longer be any more special than 36 or 49 (though I think I would still like it more than 49 since it is even, and I have an irrational dislike of odd numbers). There would be other ramifications, however. 5 would no longer be as special as it is, since it would no longer be half of the base. If we used a base-6 system, 3 would take on many of the nice properties of 5, becoming even more popular than it already is, and 5 would be relegated to the awkward position of 7 or 11 as an ungainly prime number. The reason the properties of numbers change based on the system one is working with is because of the tricks you learn to do mental math. If you are not working in a base-10 system, then multiplying by 10 no longer simply shifts things one position over and adds a zero. Instead, whatever the base of your system now does that. The entire numerical field in one's head must twist and contort to fit the new system.

The funny thing is, I have a really hard time picturing any of these ramifications, because my mind automatically works hard at translating things back into decimal. It is one of the reasons I find the binary system nicer to work with than the ternary system, since it is easier to mentally translate binary to decimal and thereby visualize what I am working with. Numbers have their properties in my head based on the decimal system, much like words have their meanings rooted very strongly in English. I might know some German words, but they are more like code words in which I have memorized their English translation rather than additional words with subtle connotations in their own right. Likewise, German grammar is an artificial system of rules that I must impose upon the sentences which I compose in my head in English. My Russian is much better, in that there are things I can successfully 'think in Russian' about, and I suppose that might happen if I were to exclusively operate in another numerical base for a while. I can even see that starting to happen with the ternary system and the Cantor Set, because I am perfectly happy to do the majority of my Cantor Set thinking in ternary. It just makes relating that to other areas of mathematics somewhat burdensome, because then I am stuck trying to mentally translate the entirely unwieldy ternary system to decimal.

Anyway, this isn't exactly the post I had envisioned to commemorate my 100th blog post, but I hope my meandering ramblings about numbers were at least vaguely interesting. The main point I was trying to make was that the appreciation for 100 that exists is mainly based on our convention of using a base-10 system rather than anything else, yet the fact that the decimal properties of numbers are a convention goes largely unacknowledged.