On Thu, Aug 30, 2012 at 4:42 PM, Peter Duveen <email@example.com> wrote: > Kirby, there is a nice method to calculate the log of a number using continued fractions. > > Say, you have 4 = 10^x. This method allows you to evolve a numerical value for x by solving that equation using continued fractions. It's a bit tedious, but interesting. Curious and thoughtful students will always wonder how they can actually calculate the log of a number from first principles. A teacher should be able to supply an answer. >
Yes, I agree. If the student is developing fluency with a language for automating algorithms, then all the tedium in working it through a first time, when writing tests and the program, pays off in the form of a running asset which may be then incorporated into additional projects.
I've had interests similar to yours in finding 2nd roots and awhile back implemented some well-known algorithm. I'm not revisiting the details tonight, although I did tweak the code for the latest Python for testing purposes.
> This method is in a mid-nineteenth century mathematics text I came across in Google Books. > > Another method that was not taught in my student days is polynomial division. Newton was familiar with the method, and uses it to expand an expression into an infinite series. I used it recently to derive a formula for the difference between two variables, each raised to the nth power, where n is an integer. The answer is something like a^n - b^n = (a - b)[a^(n-1)b + a^(n-2)b^2....+ ab^(n-2) + b^(n-1)].
I've done some work with continued fractions as well. Example: