On 09/12/2010 09:35 PM, Darren New wrote:

> And while I took and understood a great deal of math, it wasn't until
> physics class that I suddenly said "Oh, *that* is what an integral is for!"

I read my dad's "elementary calculus" textbook when I was a teenager. 
The only thing I discovered was Pythagoras' theorem (which, 
astonishingly, I had never even heard of before). The rest of the book 
made absolutely no sense to me at all. It talked a lot, and it had some 
formulas that gradually got more and more complicated, but I had no idea 
what the hell it was trying to *do*.

Fast forward a few years to a quants ("quantitative methods") class some 
day at college. (I'm not exactly sure what this has to do with 
"computing", but anyway...) We were working on graphing the income, 
profit, etc. of a fictional company. Rather than invent some fictional 
data, DKJ invented some low-degree polynomial that was supposed to 
represent income, and another that represented costs, and we were 
sitting there computing this stuff with pencil and paper.

I don't know if you've ever tried it, but tabulating even a low-degree 
polynomial by hand is tedious, even with a calculator. You tabulate the 
income polynomial, tabulate the cost polynomial, subtract one column 
from the other, and that gives you monthly profit. You then compute a 
running total of that to determine how much money in total this company 
has made or lost.

I figured out that we only actually *need* the final column. Well, you 
can subtract one polynomial from the other and do some simplifications. 
That enables you to compute the profit with fewer operations. But now 
how the hell do you compute the running total? By observing the general 
shape of the graph I was plotting (with pencil and paper, mind you), I 
was able to fit a polynomial to it. And you know, it's the damnedest 
thing... the coefficients seemed to be somehow *related* to the original 
polynomial.

At this point DKJ peered over my shoulder, and pointed out that what I 
had just done was "integral calculus", and some guy called Laplace had 
invented it 200 years ago. He showed me the general formula for 
integrating a polynomial. I was kind of suspicious at the exact integral 
coefficients; of all the possible real numbers in existence, that seems 
like a rather large coincidence. None the less, I went home and managed 
to find my dad's old calculus book.

Now it actually made perfect sense. :-P

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