
Re: Square root of six
Posted:
Aug 30, 2012 12:41 PM


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 offloading 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 selfrunning 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, sidebarred. 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

