# Existence of Dyadic Rational between two Rationals

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## Theorem

Let $a$ and $b$ be rational numbers such that $a < b$.

Then there exist integers $m$ and $r$ such that:

- $a < \dfrac m {2^r} < b$

That is, there exists a dyadic rational between any pair of rational numbers.

## Proof

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Exercise $2$