In the resource Staircase sequences you may have seen that we had a sequence of increasingly accurate rational approximations for the golden ratio \(\dfrac{1+\sqrt{5}}{2}\) and for \(\sqrt{2}\). You might have wondered how we came up with these sequences in the first place. Maybe you wondered whether it is possible to express other numbers as continued fractions.

In this problem we will explore these two key questions:

  1. Given a continued fraction, can you find the number that it represents?

  2. Given a number, can you write it as a continued fraction?

A simple continued fraction is an expression of the form \[x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \dotsb}},\]

where \(a_0\) is any integer and \(a_1, a_2, \dots\) are positive integers.

As you consider the two key questions you may like to think about some of the mini-questions below. These may be attempted in any order and might help you to get going.

Can you write \[1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4}}}\] as a more traditional fraction?

Can you express \(\frac{45}{16}\) as a continued fraction?

Can you find a continued fraction to represent \(\sqrt{3}\)?

Is it easier to find the number represented when the continued fraction is finite or infinite?

What types of numbers produce finite continued fractions? What about infinite continued fractions?

Which numbers are easier to express as continued fractions? Why?

Can every irrational number be written as a continued fraction?