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'?