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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?

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Not Continued Fractions

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

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Comparing Continued Fractions

Which of these continued fractions is bigger and why?

Good Approximations

Stage: 5 Challenge Level: Challenge Level:1

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'?