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G 30

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May 25, 2007
inductionblogsirrational numberIrrational numbers

Problem Statement

Let α=0.d1d2d3⋯\alpha=0.d_{1}d_{2}d_{3} \cdots be a decimal representation of a real number between 00 and 11. Let rr be a real number with ∣r∣<1\vert r \vert<1. [*] If α\alpha and rr are rational, must ∑i=1∞diri\sum_{i=1}^{\infty} d_{i}r^{i} be rational? [*] If ∑i=1∞diri\sum_{i=1}^{\infty} d_{i}r^{i} and rr are rational, α\alpha must be rational?
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