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## 'Good Approximations' printed from http://nrich.maths.org/

Why do this problem?
For a better understanding of rational and irrational numbers.

Possible approach
Use this problem as part of a lesson series on number to include some or all of:

- proof root 2 is irrational
- converting periodic decimals to rational numbers
- proof that every rational number has a periodic decimal expansion
- the rational numbers are countable (see Route to Infinity )
- the irrational numbers are uncountable (see the article Infinity is not a number ).

Key question

Why are the finite continued fractions which follow a regular pattern called 'convergents'?