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

In reply to Re: Empirically solving complex problems by zentara
in thread Empirically solving complex problems by oakbox

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.