View source for Mu (letter)

{{short description|Twelfth letter of the Greek alphabet}}
{{Other uses|MU (disambiguation){{!}}MU}}
{{Dist|Nu (letter)|M|Em (Cyrillic)}}
{{Redirect|µ|the micro sign|Micro-}}
{{Greek Alphabet|letter=mu}}

'''Mu''' ({{IPAc-en|ˈ|m|(|j|)|uː}};<ref>{{Cite book|title=The Chambers Dictionary|publisher=Chambers|year=2003|isbn=0-550-10105-5|edition=9th|chapter=mu}}</ref><ref>{{OED|mu}}</ref> uppercase '''Μ''', lowercase '''μ'''; [[Ancient Greek]] {{lang|grc|μῦ}} {{IPA|el|mŷː|}}, {{langx|el|μι}} or μυ—both {{IPA|el|mi|}}) is the twelfth [[Letter (alphabet)|letter]] of the [[Greek alphabet]], representing the [[voiced bilabial nasal]] {{IPA|el|m|IPA}}. In the system of [[Greek numerals]] it has a value of 40.<ref>{{Cite book |title=A Greek Grammar for Schools and Colleges |last=Hadley |first=James |location=New York |publisher=American Book |year=1884 |page=79}}</ref> Mu was derived from the [[Egyptian hieroglyph]]ic symbol for water, which had been simplified by the [[Phoenicia]]ns and named after their word for water, to become [[𐤌]]{{ref|mem|img}} (mem). Letters that derive from mu include the [[M|Roman M]] and the [[Em (Cyrillic)|Cyrillic М]], though the lowercase resembles a small [[U|Latin U]] (u).
[[File:Mu uc lc.svg|thumb|Greek letter mu]]

==Names==

===Ancient Greek===
In Greek, the name of the letter was written {{lang|grc|μῦ}} and pronounced {{IPA|[mŷː]}}.

===Modern Greek===
In [[Modern Greek]], the letter is spelled {{lang|el|μι}} and pronounced {{IPA|el|mi|}}. In polytonic orthography, it is written with an [[acute accent]]:&nbsp;{{lang|el|μί}}.<ref>{{Cite book|url=http://e-library.iep.edu.gr/iep/collection/browse/item.html?code=01-17160&tab=02&start=40#i|title = Neoelliniki Grammatiki (Tis Dimotikis)}}</ref><ref>{{Cite book|url=http://e-library.iep.edu.gr/iep/collection/browse/item.html?code=01-18549&tab=01|title = Grammatiki tis Dimotikis Glossas}}</ref>

==Use as symbol==
The lowercase letter mu (μ) is used as a special symbol in many [[academic field]]s. Uppercase mu is not used, because it appears identical to Latin  [[M]].

===Prefix for units of measurement===
"μ" is used as a [[unit prefix]] denoting a factor of 10<sup>−6</sup> (one [[millionth]]), in this context, the symbol's name is "[[Micro-|micro]]".
* [[Metric prefix]]
* [[International System of Units]] prefix, also known as "SI prefix"
* The micrometre with a symbol of "μm" can also be referred to as the non-SI term "[[micron]]".

===Mathematics===

"μ" is conventionally used to denote certain things; however, any Greek letter or other symbol may be used freely as a variable name.
* a [[measure (mathematics)|measure]] in [[measure theory]]
* [[Mu operator|minimalization]] in [[Computability theory (computer science)|computability theory]] and [[Recursion theory]]
* the [[integrating factor]] in [[ordinary differential equations]]
* the degree of [[Membership (set theory)|membership]] in a [[fuzzy set]]<ref>{{Cite book |last=Chaira |first=Tamalika |title=Fuzzy set and its extension: the intuitionistic fuzzy set |date=2019 |publisher=Wiley |isbn=978-1-119-54419-7 |location=Hoboken, NJ |page=3 |quote=If X be a collection of objects denoted by X, then a fuzzy set A in X is defined as:… where μ{{sub|A}} X is the degree of membership of X in A.}}</ref>
* the [[Möbius function]] in [[number theory]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Möbius Function |url=https://mathworld.wolfram.com/MoebiusFunction.html |access-date=2025-01-24 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite web |title=DLMF: §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory |url=https://dlmf.nist.gov/27.2 |access-date=2025-01-31 |website=dlmf.nist.gov}}</ref>
* the population [[mean]] or [[expected value]] in [[probability]] and [[statistics]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Mean |url=https://mathworld.wolfram.com/Mean.html |access-date=2025-01-24 |website=mathworld.wolfram.com |language=en}}</ref>
* the [[Ramanujan–Soldner constant]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Soldner's Constant |url=https://mathworld.wolfram.com/SoldnersConstant.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=Soldner's constant, denoted μ (or sometimes c)...}}</ref>

===Physics and engineering===
In [[classical physics]] and [[engineering]]:
* the [[friction#Coefficient of friction|coefficient of friction]]<ref>{{Cite book |last=Dante |first=Roberto C. |title=Handbook of friction materials and their applications |date=2016 |publisher=Woodhead Publishing |isbn=978-0-08-100619-1 |series=Woodhead Publishing in materials |location=Cambridge |pages=2 |quote=In the case of dry friction w(X) is the coefficient of friction μ, which is affected by temperature speed and other variables…}}</ref> (also used in aviation as braking coefficient (see [[Braking action]]))
* [[reduced mass]] in the [[two-body problem]]<ref>{{Cite book |last=Kulp |first=Christopher W. |title=Classical mechanics: a computational approach with examples using mathematica and Python |last2=Pagonis |first2=Vasilis |date=2021 |publisher=CRC Press |isbn=978-1-138-49528-9 |edition=1st |location=Boca Raton, FL |quote=The single mass ''μ'' is called the ''reduced mass'' of the two-body system and represents an effective inertial mass appearing in the two-body problem.}}</ref>
* [[Standard gravitational parameter]] in celestial mechanics<ref>{{Cite book |last=Dodd |first=Richard |title=Using SI units in astronomy |date=2012 |publisher=Cambridge University Press |isbn=978-0-521-76917-4 |location=Cambridge |pages=96 |quote=The standard gravitational parameter μ, is the product of the Newtonian gravitational constant G, and the mass of a given celestial body.}}</ref>
* [[linear density]], or mass per unit length, in strings and other one-dimensional objects<ref>{{Cite book |last=Walker |first=Jearl |title=Halliday & Resnick fundamentals of physics |last2=Resnick |first2=Robert |last3=Halliday |first3=David |date=2014 |publisher=Wiley |isbn=978-1-118-23376-4 |edition=10th |location=Hoboken, NJ |pages=452 |quote=We call this ratio the ''linear density μ'' of the striing.}}</ref>
* [[permeability (electromagnetism)|permeability]] in electromagnetism<ref>{{Cite book |last=Jiles |first=David |title=Introduction to magnetism and magnetic materials |date=2016 |publisher=CRC Press, Taylor & Francis Group |isbn=978-1-138-44149-1 |edition=Third |location=Boca Raton London New York |pages=42 |quote=We can now make a general statement for the permeability ''μ'' and susceptibility ''{{mvar|χ}}''}}</ref>
* the [[magnetic dipole moment]] of a [[Electric current|current]]-carrying [[Coil (electrical engineering)|coil]]
* [[Viscosity#Dynamic (shear) viscosity|dynamic viscosity]] in fluid mechanics
* the [[Amplifier|amplification]] factor or [[Gain (electronics)|voltage gain]] of a [[triode]] [[vacuum tube]]<ref>{{cite book |last=Ballou |first=Glen|title=Handbook for Sound Engineers: The New Audio Cyclopedia |publisher=Howard W. Sams Co. |year=1987 |edition=1 |page=250 |isbn=0-672-21983-2 |quote=''Amplification factor or voltage gain'' is the amount the signal at the control grid is increased in amplitude after passing through the tube, which is also referred to as the Greek letter μ (mu) or voltage gain (V<sub>g</sub>) of the tube.}}</ref>
* the [[electrical mobility]] of a charged particle
* the [[Slowed rotor#Theory|rotor advance ratio]], the ratio of aircraft [[airspeed]] to rotor-tip speed in [[rotorcraft]]<ref>"[http://rotorcraft.arc.nasa.gov/tutorial/Nomenclature.htm#mu Nomenclature]" ''NASA''</ref><ref>{{Cite web|url=http://dictionary.reference.com/browse/advance+ratio|title=Definition}}</ref><!--large download <ref>Oliver, Michael. "[http://dspace.mit.edu/bitstream/handle/1721.1/28910/60495046.pdf;jsessionid=4914214DD2723B08238E70B668D773DC?sequence=1 A parametric analysis of the start-up procedure and flight characteristiscs of a gliding autogyro]" page 43. ''[[Massachusetts Institute of Technology School of Engineering#Aeronautics and Astronautics|MIT Aero]]'', February 2005. Accessed: December 2013. Size: 15MB and 114 pages</ref> -->
* the [[pore water pressure]] in saturated soil

In [[particle physics]]:
* the elementary particles called the [[muon]] and antimuon
* the [[proton-to-electron mass ratio]]<ref>{{Cite journal |last=Thompson |first=Rodger I. |date=2017 |title=Fundamental constants as monitors of the universe |url=http://www.worldscientific.com/doi/abs/10.1142/9789813226609_0476 |journal=MG14 |language=en |publisher=WORLD SCIENTIFIC |pages=3657–3663 |doi=10.1142/9789813226609_0476 |isbn=978-981-322-659-3 |quote=There are now observational constraints on the time variation of the proton to electron mass ratio μ at the 10−7 level.|arxiv=1602.03192 }}</ref>

In [[thermodynamics]]:
* the [[chemical potential]] of a system or component of a system

=== Computer science ===
In [[evolutionary algorithm]]s:
* μ, population size from which in each generation λ offspring will generate (the terms μ and λ originate from [[evolution strategy]] notation)

In [[type theory]]:
* Used to introduce a [[recursive data type]].  For example, <math>\text{list}(\tau) = \mu{}\alpha{}.1 + \tau{}\alpha</math> is the type of lists with elements of type <math>\tau</math> (a [[type variable]]):  a [[Sum type|sum]] of [[Unit type|unit]], representing {{mono|nil}}, with a pair of a <math>\tau</math> and another <math>\text{list}(\tau)</math> (represented by <math>\alpha</math>).  In this notation, <math>\mu</math> is a binding form, where the variable (<math>\alpha</math>) introduced by <math>\mu</math> is bound within the following term (<math>1 + \tau{}\alpha</math>) to the term itself.  Via substitution and arithmetic, the type expands to <math> 1 + \tau + \tau^2 + \tau^3 + \cdots</math>, an infinite sum of ever-increasing [[Product type|products]] of <math>\tau</math> (that is, a <math>\tau{}\text{ list}</math> is any <math>k</math>-tuple of values of type <math>\tau</math> for any <math>k \ge 0</math>).  Another way to express the same type is <math>\text{list}(\tau) = 1 + \tau{}\text{list}(\tau)</math>.

===Chemistry===
In [[chemistry]]:
* the prefix given in [[IUPAC]] nomenclature for a [[bridging ligand]]

===Biology===
In [[biology]]:
* the [[mutation rate]] in population genetics
* A class of [[Immunoglobulin heavy chain]] that defines [[Immunoglobulin M|IgM]] type [[Antibody|Antibodies]]

===Pharmacology===
In [[pharmacology]]:
* an important [[opiate receptor]]

===Orbital mechanics===
In [[orbital mechanics]]:
* [[Standard gravitational parameter]] of a [[celestial body]], the product of the [[gravitational constant]] ''G'' and the [[mass]] ''M''
* planetary discriminant, represents an experimental measure of the actual degree of [[Clearing the neighbourhood|cleanliness]] of the orbital zone, a [[wikt:criterion|criterion]] for [[IAU definition of planet|defining]] a [[planet]]. The value of μ is calculated by dividing the mass of the candidate body by the total mass of the other objects that share its orbital zone.

===Music===
* [[Steely Dan|Mu chord]]
* Electronic musician [[Mike Paradinas]] runs the label [[Planet Mu]] which utilizes the letter as its logo, and releases music under the pseudonym μ-Ziq, pronounced "music"
* Used as the name of the school idol group μ's, pronounced "muse", consisting of nine singing idols in the anime [[Love Live! School Idol Project]]
* Official [[fandom name]] of Kpop group [[F(x) (band)|f(x)]], appearing as either '''MeU''' or 'μ'
* Hip-hop artist Muonboy has taken inspiration from the particle for his stage name and his first EP named Mu uses the letter as its title.

===Cameras===
The [[Olympus Corporation]] manufactures a series of digital cameras called Olympus μ {{IPA|[mju:]}}<ref>{{cite web |url=http://www.olympus-global.com/en/corc/history/camera/m_series/ |title=Olympus History : μ[mju:] (Stylus) Series}}</ref> (known as [[Olympus Stylus Epic|Olympus Stylus]] in North America).

===Linguistics===
In [[phonology]]:
* [[mora (linguistics)|mora]]

In [[syntax]]:
* μP (mu phrase) can be used as the name for a functional projection.<ref>{{cite journal|last=Johnson|first=Kyle|title=Object Positions|journal=Natural Language and Linguistic Theory|year=1991|volume=9|pages=577–636|doi=10.1007/BF00134751|issue=4|s2cid=189901613}}</ref>

In [[celtic languages|Celtic]] linguistics:
* /μ/ can represent an [[Old Irish]] nasalized labial [[fricative]] of uncertain articulation, the ancestor of the sound represented by [[irish language|Modern Irish]] ''mh''.

==Unicode==

The lowercase mu (as "micro sign") appeared at {{tt|0xB5}} in the 8-bit [[ISO-8859-1]] encoding, from which Unicode and many other encodings inherited it. It was also at {{tt|0xE6}} in the popular [[CP437]] on the IBM PC. Unicode has declared that a "real" mu is different than the micro sign.<ref>Unicode Code Charts: [https://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]</ref>{{Failed verification|date=December 2024}}

* {{unichar|00B5|html=}}
* {{unichar|039C|html=}}
* {{unichar|03BC|html=}}
* {{unichar|2C98|html=}}
* {{unichar|2C99|html=}}
* {{unichar|1D6B3|html=}}{{efn|The {{sc|mathematical}} symbols are only for math. Stylized text should use markup.}}
* {{unichar|1D6CD|html=}}
* {{unichar|1D6ED|html=}}
* {{unichar|1D707|html=}}
* {{unichar|1D727|html=}}
* {{unichar|1D741|html=}}
* {{unichar|1D761|html=}}
* {{unichar|1D77B|html=}}
* {{unichar|1D79B|html=}}
* {{unichar|1D7B5|html=}}
{{notelist}}

==See also==
{{Wiktionary|Μ|μ}}
* [[Greek letters used in mathematics, science, and engineering]]
* [[Fraser alphabet#Consonants]]

==References==
{{Reflist}}

[[Category:Greek letters]]