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The Continued Fraction Expansion of 1 + sqrt(2)

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with side s and diagonal d then performing a division algorithm (as mentioned in the prior question), on d+s,s yields an infinite sequence of 2's. what can be concluded about the ratio of the diagonal of the square to the side? what real number do you think an infinite sequence of 1's corresponds to?

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Solution Summary

Here we show that the number with all continued fraction coefficients equal to 2 is 1 + sqrt(2).

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We know by the Pythagorean theorem that d/s is equal to sqrt(2), so (d+s)/s = 1 + sqrt(2). Thus it suffices to show that 1 + sqrt(2) = 2 + 1/(2 + 1/(2 + 1/(2 + ..., i.e. that all the continued fraction ...

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