### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

### Comparing Continued Fractions

Which of these continued fractions is bigger and why?

# Good Approximations

##### Age 16 to 18Challenge Level

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'?