Re: Empirically solving complex problems

He graphed a function on the board and started working out how to find the area below the curve. As an aside, he started to explain that a chemist, confronted with the same problem, would simply draw the graph out 10 times and weigh the resulting graphs to arrive at the answer.

Hmm that's interesting, because it ininsinuates that the technique can't be done in mathematics...but it can, and it is the basis of calculus...limits.

So how did the "first mathemeticians", who didn't have the calculus formulas, figure out the area under the curve? Well, they did it just like the chemists. He divided the area under the curve into tall thin rectangles, of width x, and took the average value of the 2 y values at the top, got the area of the rectangles, and summed them up.

So then someone asked, "how can we get more accuracy?". They figured out as you narrowed the x width, the average y value at the top of the rectangle, "converged" to the same value. THE LIMIT.

It didn't take long for them to come up with the dx and dy notations, and to put it into formulas.

So even today, there is a whole branch of engineering called "applied numerical analysis", where you write programs exactly as described above. If your equation, or conditions, are so complex, that they can't be put into a simple formula, you break it up into intervals, either spatial or time, and compute the value of each interval separately, then sum up the results. Computers are so fast, it is usually the fastest way to model something.

I was in EE, loved the math, but dropped out to ponder the "meaning of it all" :-) Check out this story about how some people actually can see numbers as shapes....it makes you wonder about how puny our minds really are. Math Genius

I'm not really a human, but I play one on earth. flash japh

Comment on Re: Empirically solving complex problems

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Re^2: Empirically solving complex problems by bageler (Hermit) on Mar 05, 2005 at 16:54 UTC
I was in EE, loved the math, but dropped out to ponder the "meaning of it all" funny, I dropped out of EE to sell porn on the internet too.	[reply]
Re^2: Empirically solving complex problems by fizbin (Chaplain) on Mar 07, 2005 at 06:19 UTC
So how did the "first mathemeticians", who didn't have the calculus formulas, figure out the area under the curve? Well, they did it just like the chemists. He divided the area under the curve into tall thin rectangles, of width x, and took the average value of the 2 y values at the top, got the area of the rectangles, and summed them up. While conceptually I have no problem with this explanation, please do not confuse the concept with the historical approaches to finding the area inside a curve. `-- @/=map{[/./g]}qw/.h_nJ Xapou cets krht ele_ r_ra/; map{y/X_/\n /;print}map{pop@$_}@/for@/` [download]	[reply] [d/l]
Re^3: Empirically solving complex problems by zentara (Cardinal) on Mar 07, 2005 at 13:21 UTC
Yeah, that just goes to show, how "our world vision" is shaped by our educational processes. In my school, the origin began with DeCartes(with only footnotes to the Greeks). I'm not really a human, but I play one on earth. flash japh	[reply]