On 09/12/2010 07:50 PM, Warp wrote:

>    That reminds me vividly of something which I have noticed and realized
> all by myself: I have never seen *anywhere* the *logical* explanation of
> how you can easily calculate, for example, the sum of all the integers
> from 1 to 100. It's *always* just the raw and sterile formula, and that's
> it. No explanation, no intuitive nor logical way of deducing it. Just the
> formula and that's it.
>
>    The raw formula is rather useless by itself when you don't understand
> where it's coming from.

I do recall seeing in my sister's maths book the infamous quadratic 
solution formula. The book then went on to explain how to derive it...

...by starting with the quadratic equation and applying a seemingly 
arbitrary sequence of exotic algebraic manipulations to it. Sure enough, 
the final result *is* the standard formula. But surely no person would 
have thought of plucking this exact random series of transformations out 
of thin air.

In other words, it demonstrates that the formula *works*, but offers no 
intuitive insight into *why* it works. I got /that/ a bit later that 
day, after I discovered something called "factorisation"...

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