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Farey Approximation

Stage: 5 Short Challenge Level: Challenge Level:1

Why do this problem?
Finding numerical approximations is a useful skill. This problem introduces a method for finding rational approximations to irrational numbers as an application of mediants. It also links to a set of problems involving Farey Sequences and infinite sets of circles called Ford Circles so that learners get a glimpse of the broader picture of interconnectedness of mathematical concepts.

Possible approach
A spreadsheet can be used to assist with the calculations. If a calculator is used it helps to set out the working in a table. If learners work in pairs they can discuss which interval to choose at each stage and check their results with each other.

Possible extension
Read the articles Continued Fractions 1 and Approximations, Euclid's Algorithm and Continued Fractions which explain the use of continued fractions to give rational approximations to irrational numbers. The continued fraction method arrives at an approximation to $\pi$ of $$ \pi \approx 3+ {1\over\displaystyle 7 + {1\over \displaystyle 15.9966}}.$$ This gives the approximation (which is accurate to 6 decimal places): $$\pi \approx 3+ {1\over \displaystyle 7 + {1\over \displaystyle 16}} = {3 + {16\over 113}} = {355\over 113}.$$ The 7 corresponds to the 7 steps involving the Farey intervals with lower end $3$ and the 16 corresponds to 16 steps involving Farey intervals with upper end $\frac{22}{7}$. Repeating this algorithm we have $$\pi \approx 3+ {1\over \displaystyle 7 + {1\over \displaystyle 15 + {1\over \displaystyle 1 + {1\over \displaystyle 293 + {1\over \displaystyle 10.320556}}}}}$$ and so on ...

By analogy the steps for $\sqrt 2$ in the Farey sequence approximation process are repeated 2,2,2,2,2,2,... times which gives the continued fraction approximation, accurate to 4 decimal places:
$$\sqrt 2 \approx 1+ {1\over \displaystyle 2 + {1\over \displaystyle 2 + {1\over \displaystyle 2 + {1\over \displaystyle 2 + {1\over \displaystyle 2+{1\over \displaystyle 2}}}}}}=\frac{239}{169}$$ and so on ....

Possible support
First do an example by interval halving as this involves choice of the appropriate interval at each stage in a similar way to the method using the Farey mediant.