A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.
Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
Which of these continued fractions is bigger and why?
Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
An article introducing continued fractions with some simple puzzles for the reader.
Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
Can you tangle yourself up and reach any fraction?
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
It would be nice to have a strategy for disentangling any tangled ropes...
In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.
A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
This article sets some puzzles and describes how Euclid's algorithm and continued fractions are related.
Find the equation from which to calculate the resistance of an infinite network of resistances.
Use Euclid's algorithm to get a rational approximation to the number of major thirds in an octave.
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!