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Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

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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?

Golden Fractions

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Some very pretty results here! It should come as no surprise to see Fibonacci numbers linked with the Golden Ratio. This is the simplest of all infinite continued fractions and the sequence of approximants converges very rapidly to $\phi -1$.