### Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

### There's a Limit

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?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

# Euclid's Algorithm and Musical Intervals

##### Age 16 to 18Challenge Level

See the problem Tuning and Ratio in which you have to find a decimal approximation for this ratio using logarithms. Here you must find an approximation in the form of a ratio of two integers without using logarithms.

You could use the same method as given for finding rational approximations to $\pi$ in the article Approximations, Euclid's Algorithm and Continued Fractions.