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Dyadic fraction

A dyadic fraction is a fraction in which the denominator is a power of two, i.e. a rational number of the form a/2b where a is an integer and b is a natural number. (Like fractions of an inch as commonly used in the US, for instance) These are precisely the numbers whose binary expansion is finite. The set of all dyadic fractions is dense in the real line; it is a rather "small" dense set, which is why it sometimes occurs in proofs, see for instance Urysohn's lemma. The dyadic fractions form a subring of Q.

what properties does this ring have?

The surreal numbers are generated by an iterated construction principle which starts by generating all finite dyadic fractions, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.

The ancient Egyptians used Horus-eye notation for dyadic fractions.