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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 18
Challenge 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'?