# Arithmetic with Continued Fractions

**Length:** 60 minutes

## Description

Multiprecision arithmetic algorithms usually represent real numbers
as decimals, or perhaps as their base-2^{n} analogues. But
this representation has some puzzling properties. For example, there
is no exact representation of even as simple a number as
one-third. Continued fractions are a practical but little-known
alternative.

Continued fractions are a representation of the real numbers that
are in many ways more mathematically natural than the usual decimal or
binary representations. All rational numbers have simple
representations, and so do many irrational numbers, such as sqrt(2) and
*e*^{1}. One reason that continued fractions are not
often used, however, is that it's not clear how to involve them in
basic operations like addition and multiplication. This was an
unsolved problem until 1972, when Bill Gosper found practical
algorithms for continued fraction arithmetic.

In this talk, I explain what continued fractions are and why they
are interesting, how to represent them in computer programs, and how
to calculate with them.

## Complete Slides

- Arithmetic with Continued Fractions
- What we'll do
- How do we represent real numbers for calculation?
- Decimal representation
- Decimal arithmetic
- Continued Fractions
- Truncation
- Irrational numbers
- Continued fractions
- Change of stance
- Why not continued fractions?
- The catch
- Continued fraction objects
- Input
- Output
- When to output?
- An example
- Let's ask for a term from _z_
- Aha!
- [I[ex1+2-11-7]I]
- Details
- This time with feeling
- Continued fraction arithmetic
- When to output?
- Arithmetic
- Implementation
- Thank you
- Bonus slides and deleted material
- Important properties of continued fractions
- Comparing continued fractions is easy
- Truncation

Complete slides and illustrations for the talk.

## Additional Materials

A small implementation of the Gosper algorithm in C.

Gosper's HAKMEM notes, his
longer monograph, and some other papers and materials.

Return to:
Universe of Discourse main page |
Perl Paraphernalia |
Other Classes and Talks

mjd-perl-yak+@plover.com