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Topic: Square root of six
Replies: 21   Last Post: Aug 31, 2012 12:47 PM

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kirby urner

Posts: 1,704
Registered: 11/29/05
Re: Square root of six
Posted: Aug 30, 2012 12:41 PM
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On Thu, Aug 30, 2012 at 7:03 AM, Peter Duveen <pduveen@yahoo.com> wrote:
> So my point, I guess, is that, for the purposes of teaching, there is much to be gained from a knowledge of the historical sequence of mathematical development.


I completely agree with this.

The Egyptian style of expressing fractions appears to have had
everything to do with apportionment of grains through containers of
fixed volume. Along the Nile, with fields but a thin margin on either
side, such apportionment was life itself (its circulatory system).

These days, efficiently packing a truck and routing it, such that
packages in need of off-loading aren't buried in the back. UPS trucks
have shelves for random access right? That way routing and packing
stay two different problems. But some loads are too bulky for
shelving.

Continued fractions are another area that, along with much of number
theory, has fallen by the wayside in a youth's education. Yet these
are the ideal little challenges one looks for when learning a
computer language. More fun than inverting a matrix, or in the same
category at least, with opportunities to use recursion. You get to
scrunch up your brain and do something precise. Then, unlike with
paper and pencil, you actually get a self-running algorithm more often
than not. Something you wrote. Something you might reuse.

Of course with just your plain old everyday interactive prompt you can
start playing with these fractions:

>>> 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/1)))))))))))))))))))))))
1.6180339886704433

That looks tedious to type but there's cut and pasting involved, plus
the sensor visually shows whether your parentheses are balanced, so no
great feat of concentration was required. And look: approaching phi
(there'd need to be more of a proof).

A great story line to follow is, of course, the evolution of
cryptography. The role of computers in both decrypting and encrypting
becomes a focus, along with a bevy of little problem solving
challenges, such as how to implement this or that algorithm.

http://4dsolutions.net/ocn/crypto0.html (some of my own writings on
the subject)

One need not go whole hog, thinking this is a college course on
nothing but crypto. Nor must the focus stay so heads down with
blinders, nose in the exercises.

This ain't your grandpappy's math, wherein any history was verboten,
cordoned off, side-barred. History is often front and center, such as
when we discuss the history of tabulation (form clay tablets to
Hollerith machines to SQL), or the history of glyph representation
(from clay tablets though unicode).

Kirby



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