As I perused some of the other WordPress bloggers’ blogs, I began to realize a fundamental quality that my own blog lacked: content. One would expect that I would have realized this sooner, but apparently I was blinded by my lust for unending, nonsensical rants. But once I’d seen the light, that got me wondering; “What the hell have I got that’s worth talking about.” After several hours and many false starts, I realized I had at least one thing: mathematics.
Now, I do admit that I’m something of an outsider, even in mathematical circles. I’ve never been incredibly fond of the globs of terminology my fellow math-heads throw around (but that may be my ego doing the talking). Still, I think there’s something to be gained through a complete ignorance of all these extra words. (But I may be wrong: You are talking to a guy who didn’t take algebra until 9th grade, and hated math until he was about fifteen.)
Now, for those of you who slogged through all of that, here comes the mathematics:
NOTE: The moment I finished this article, I ran across the article on Wikipedia about Donald Knuth’s “Up-Arrow Notation.” It’s pretty much exactly the same as my n-multiplications. (I can hear all you mathematicians snickering at me…)
Sometime last semester, I got thinking about the different kinds of multiplication-like operations that could be performed, and I came up with a list of the “fundamental multiplications” (I now call them n-multiplications):
m_0(a, b) = 1
m_1(a, b) = a + b
m_2(a, b) = a * b
m_3(a, b) = a ^ b
I found the notebook I’d written this list in the other day, and it got me thinking: wouldn’t it be possible to write the higher-order multiplications in terms of the lower-order ones:
m_2(a, b) = a * b = a + a + a + … + a (b additions) = m_1(a, m_1(a, m_1(a, … a))) (b repeats)
And furthermore, you could write m3(a, b) thus:
m_3(a, b) = a ^ b = a * a * a * … * a (b multiplications) = m_2(a, m_2(a, m_2(a, … a)))
So, I took a bit of a leap, and in order to extend the idea of multiplications, I define the “property of reduction” for any multiplication m_n(a, b):
m_n(a, b) = m_(n-1) (a, m_(n-1)(a, … a)))
That’s about as far as it’s gone so far. If you really feel the need to correct me or add to the idea, e-mail the idea to firstname.lastname@example.org.
And yes, I’m aware that not all the n-multiplications are symmetric or associative. That’s one of those bits that remains to be added.