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'?