Editing Dyadic rational (section)

==Additional properties==
[[File:Dyadic sqrt2 approximation.svg|thumb|Dyadic rational approximations to the [[square root of 2]] (<math>\sqrt{2}\approx 1.4142</math>), found by rounding to the nearest smaller integer multiple of <math>1/2^i</math> for <math>i=0,1,2,\dots</math> The height of the pink region above each approximation is its error.]]
[[File:Bad dyadic approximation.svg|thumb|upright=1.2|Real numbers with no unusually-accurate dyadic rational approximations. The red circles surround numbers that are approximated within error <math>\tfrac16/2^i</math> by <math>n/2^i</math>. For numbers in the fractal [[Cantor set]] outside the circles, all dyadic rational approximations have larger errors.]]
Every integer, and every [[half-integer]], is a dyadic rational.{{r|sabin}} They both meet the definition of being an integer divided by a power of two: every integer is an integer divided by one (the zeroth power of two), and every half-integer is an integer divided by two.

Every [[real number]] can be arbitrarily closely approximated by dyadic rationals. In particular, for a real number <math>x</math>, consider the dyadic rationals of the form {{nowrap|<math display=inline>\lfloor 2^i x \rfloor / 2^i</math>,}} where <math>i</math> can be any integer and <math>\lfloor\dots\rfloor</math> denotes the [[floor function]] that rounds its argument down to an integer. These numbers approximate <math>x</math> from below to within an error of <math>1/2^i</math>, which can be made arbitrarily small by choosing <math>i</math> to be arbitrarily large. For a [[fractal]] subset of the real numbers, this error bound is within a constant factor of optimal: for these numbers, there is no approximation <math>n/2^i</math> with error smaller than a constant times <math>1/2^i</math>.<ref>More precisely, for small positive values of <math>\varepsilon</math>, the set of real numbers that have no approximation <math>n/2^i</math> with error smaller than a constant times <math>\varepsilon/2^i</math> forms a [[Cantor set]] whose [[Hausdorff dimension]], as a function of <math>\varepsilon</math>, goes to one as <math>\varepsilon</math> approaches zero. The illustration shows this set for <math>\varepsilon=\tfrac16</math>.</ref>{{r|nilsson}} The existence of accurate dyadic approximations can be expressed by saying that the set of all dyadic rationals is [[dense set|dense]] in the [[real line]].{{r|sabin}} More strongly, this set is uniformly dense, in the sense that the dyadic rationals with denominator <math>2^i</math> are uniformly spaced on the real line.{{r|ko}}

The dyadic rationals are precisely those numbers possessing finite [[binary number|binary expansions]].{{r|ko}} Their binary expansions are not unique; there is one finite and one infinite representation of each dyadic rational other than 0 (ignoring terminal 0s). For example, 0.11<sub>2</sub> = 0.10111...<sub>2</sub>, giving two different representations for 3/4.{{r|ko|kac}} The dyadic rationals are the only numbers whose binary expansions are not unique.{{r|ko}}