Editing Dyadic rational (section)

==Applications==
===In measurement===
{{CSS image crop|Image=Kitchen weights metric imperial.jpg|bSize=427|cWidth=256|oLeft=165|oTop=27|cHeight=256|Description=Kitchen weights measuring dyadic fractions of a [[Pound (mass)|pound]] from {{nowrap|2 lb}} down to 1/64 lb (1/4 oz)|Alt=Photo of metal disks used as kitchen weights}}
Many traditional systems of weights and measures are based on the idea of repeated halving, which produces dyadic rationals when measuring fractional amounts of units. The [[inch]] is customarily subdivided in dyadic rationals rather than using a decimal subdivision.{{r|rudman}} The customary divisions of the [[gallon]] into half-gallons, [[quart]]s, [[pint]]s, and [[Cup (unit)|cups]] are also dyadic.{{r|barnes}} The ancient Egyptians used dyadic rationals in measurement, with denominators up to 64.{{r|curtis}} Similarly, systems of weights from the [[Indus Valley civilisation]] are for the most part based on repeated halving; anthropologist Heather M.-L. Miller writes that "halving is a relatively simple operation with beam balances, which is likely why so many weight systems of this time period used binary systems".{{r|miller}}

===In computing===
Dyadic rationals are central to [[computer science]] as a type of fractional number that many computers can manipulate directly.{{r|reswel}} In particular, as a data type used by computers, [[Floating-point arithmetic|floating-point numbers]] are often defined as integers multiplied by positive or negative powers of two. The numbers that can be represented precisely in a floating-point format, such as the [[IEEE floating point|IEEE floating-point datatypes]], are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals.{{r|kirk-hwu}} The same is true for [[fixed-point arithmetic|fixed-point datatypes]], which also use powers of two implicitly in the majority of cases.{{r|kneusel}} Because of the simplicity of computing with dyadic rationals, they are also used for exact real computing using [[interval arithmetic]],{{r|vdh}} and are central to some theoretical models of [[computable number]]s.{{r|ko|zr|asz}}

Generating a [[random variable]] from random bits, in a fixed amount of time, is possible only when the variable has finitely many outcomes whose probabilities are all dyadic rational numbers. For random variables whose probabilities are not dyadic, it is necessary either to approximate their probabilities by dyadic rationals, or to use a random generation process whose time is itself random and unbounded.{{r|jvv}}

===In music===
{{Image frame|content=<score sound="1"> { \new PianoStaff << \new Staff \relative c'' { \set Staff.midiInstrument = #"violin" \clef treble \tempo 8 = 126 \time 3/16 r16 <d c a fis d>\f-! r16\fermata | \time 2/16 r <d c a fis d>-! \time 3/16 r <d c a fis d>8-! | r16 <d c a fis d>8-! | \time 2/8 <d c a fis>16-! <e c bes g>->-![ <cis b aes f>-! <c a fis ees>-!] } \new Staff \relative c { \set Staff.midiInstrument = #"violin" \clef bass \time 3/16 d,16-! <bes'' ees,>-! r\fermata | \time 2/16 <d,, d,>-! <bes'' ees,>-! | \time 3/16 d16-! <ees cis>8-! | r16 <ees cis>8-! | \time 2/8 d16\sf-! <ees cis>-!->[ <d c>-! <d c>-!] } >> } </score>|caption=Five bars from [[Igor Stravinski]]'s ''[[The Rite of Spring]]''<br/>showing time signatures {{music|time|3|16}}, {{music|time|2|16}}, {{music|time|3|16}}, and {{music|time|2|8}}}}
[[Time signature]]s in Western [[musical notation]] traditionally are written in a form resembling fractions (for example: {{music|time|2|2}}, {{music|time|4|4}}, or {{music|time|6|8}}),{{r|jon-pea}} although the horizontal line of the musical staff that separates the top and bottom number is usually omitted when writing the signature separately from its staff. As fractions they are generally dyadic,{{r|libbey}} although [[Time signature#Irrational meters|non-dyadic time signatures]] have also been used.{{r|yanakiev}} The numeric value of the signature, interpreted as a fraction, describes the length of a measure as a fraction of a [[whole note]]. Its numerator describes the number of beats per measure, and the denominator describes the length of each beat.{{r|jon-pea|libbey}}

===In mathematics education===
In theories of childhood development of the concept of a fraction based on the work of [[Jean Piaget]], fractional numbers arising from halving and repeated halving are among the earliest forms of fractions to develop.{{r|hie-ton}} This stage of development of the concept of fractions has been called "algorithmic halving".{{r|pot-saw}} Addition and subtraction of these numbers can be performed in steps that only involve doubling, halving, adding, and subtracting integers. In contrast, addition and subtraction of more general fractions involves integer multiplication and factorization to reach a common denominator. Therefore, dyadic fractions can be easier for students to calculate with than more general fractions.{{r|wells}}