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How do I find a rational number in between?

The expression \frac{m}{2^n} is a terminating binary fraction, so let's express our other two fractions in binary until they differ. How do we do that, you might ask? Easy! Multiply by two, extract the Integer part, rinse and repeat. This leads to the following pair of sequences: 1 rem 8/10 & 1 rem 9/11 1 rem 6/10 & 1 rem 7/11 1 rem 2/10 & 1 rem 3/11 rem 4/10 & rem 6/11 rem 8/10 & 1 rem 1/11 Excellent, we have a difference in the fifth binary digit. We now know that \frac9{10}