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{{Short description|Seventeenth letter of the Greek alphabet}}
{{Other uses}}{{Distinguish|text=[[P]] or [[P|P (lowercase)]]}}{{Greek Alphabet|letter=rho}}
'''Rho''' ({{IPAc-en|'|r|oʊ}}; uppercase '''Ρ''', lowercase '''ρ''' or '''{{not a typo|ϱ}}'''; {{langx|el|ρο}} or {{langx|el|ρω|label=none}}) is the seventeenth letter of the [[Greek alphabet]]. In the system of [[Greek numerals]] it has a value of 100. It is derived from [[Phoenician alphabet|Phoenician letter]] [[Resh|res]] [[Image:Phoenician res.svg|20px]]. Its uppercase form uses the same [[glyph]], Ρ, as the distinct [[Latin alphabet|Latin]] letter [[P]]; the two letters have [[#Character encodings|different Unicode encodings]].

== Uses ==
[[File:NAMA Alphabet grec.jpg|thumb|The Greek alphabet on a [[black figure]] vessel, with an R-shaped rho.]]

===Greek===
Rho is classed as a [[liquid consonant]] (together with [[Lambda (letter)|Lambda]] and sometimes the nasals [[Mu (letter)|Mu]] and [[Nu (letter)|Nu]]), which has important implications for [[Morphology (linguistics)|morphology]]. In both [[Ancient Greek|Ancient]] and [[Modern Greek]], it represents an [[alveolar trill]] {{IPA|el|r|IPA}}, [[alveolar tap]] {{IPA|el|ɾ|IPA}}, or [[alveolar approximant]] {{IPA|el|ɹ|IPA}}.

In [[polytonic orthography]], a rho at the beginning of a word is almost always written ⟨{{lang|grc|ῥ}}⟩ (''rh'') with a [[rough breathing]] mark, indicating that it is [[voiceless consonant|voiceless]]. Very rarely, it is written ⟨{{lang|grc|ῤ}}⟩ (''r'') with a [[smooth breathing]] mark, indicating that it is [[voiced consonant|voiced]], instead. Rho is not written with breathing marks at any other place in a word, where it is always voiced, with the exception of double rho, which was traditionally written ⟨{{lang|grc|ῤῥ}}⟩ (''rrh''), with a smooth breathing mark over the first rho, and a rough breathing mark over the second, representing a [[gemination|geminated]] voiceless consonant. However, this practice fell out of use over the 19th century in favour of ⟨{{lang|grc|ρρ}}⟩, since double rho cannot take any other combination of breathing marks. Various Greek-derived English words containing ''rh'' and ''rrh'' derive from words containing ⟨{{lang|grc|ῤ}}⟩ and ⟨{{lang|grc|ρρ}}⟩.

The name of the letter is written in Greek as {{lang|grc|ῥῶ}} (polytonic) or ρω/[[wikt:ρο|ρο]] (monotonic).

===Other alphabets===
Letters that arose from rho include Roman [[R]] and Cyrillic [[Er (Cyrillic)|Er]] (Р).

=== Mathematics and science ===
The characters '''ρ''' and '''{{not a typo|ϱ}}''' are also conventionally used outside the Greek alphabetical context in [[science]] and [[mathematics]].
*In the [[physical sciences]] to represent:
**Densities: [[mass density]], [[air density]] or [[charge density]] (''ρ'')
**[[Resistivity]] (''ρ'')
**[[Rho meson]] (ρ<sup>+</sup>, ρ<sup>−</sup>, ρ<sup>0</sup>)
**[[Density matrix|General quantum states]]
**[[Hammett Equation]], ρ is used to represent the reaction constant, this is independent of the position and nature of the substituents of the benzene ring.
*In [[mathematics]] to represent:
**A length coordinate in [[Polar coordinate system#Conventions|polar]], [[Cylindrical coordinate system|cylindrical]], [[Spherical coordinate system|spherical]],  and [[Toroidal coordinates#An alternative separation|toroidal]] coordinate systems, and [[toroidal and poloidal]] coordinates of the Earth's magnetic field.<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
**The [[Pearson product-moment correlation coefficient|correlation coefficient]] of a population parameter
**The [[spectral radius]] of a matrix <math>A</math> denoted as <math>\rho(A)</math><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Spectral Radius |url=https://mathworld.wolfram.com/SpectralRadius.html |access-date=2025-01-22 |website=mathworld.wolfram.com |language=en}}</ref>
**The [[plastic ratio]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Plastic Constant |url=https://mathworld.wolfram.com/PlasticConstant.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=The plastic constant ρ, sometimes also called le nombre radiant, the minimal Pisot number, plastic number, plastic ratio,...}}</ref>
**The [[prime constant]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Prime Constant |url=https://mathworld.wolfram.com/PrimeConstant.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>
**The [[Greeks (finance)#Rho|sensitivity]] to interest rates of a pricing function
**The expected return of a given policy (<math>\pi</math>) in [[reinforcement learning]], denoted <math>\rho^\pi</math>
**The [[Relational algebra#Rename (ρ)|rename operation]] in relational algebra
*In [[economics]] to represent the discount rate of future cash flows{{Citation needed|date=January 2025}}
*In [[molecular biology]] to represent the [[Rho factor|Rho protein]] responsible for termination of RNA synthesis. In such occasions, it is often represented as {{unichar|03f1|Greek rho symbol}}, to avoid confusion with the Latin letter ''p''
*In [[molecular biology]] to represent the [[Rho family of GTPases]], important for cytoskeletal dynamic regulation.
*In [[ecology]] to represent the population damping ratio where ρ = λ1 / |λ2|.
*In [[computer programming]]
**The lower-case rho "⍴" means reshape in the [[APL programming language]], and by extension also queries shape
**The lower-case rho "ρ" is used for the unary [[Rename (relational algebra)|rename operation]] in [[relational algebra]]
*In [[statistics]] to represent [[Spearman's rank correlation coefficient]], commonly known as Spearman's rho<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Spearman Rank Correlation Coefficient |url=https://mathworld.wolfram.com/SpearmanRankCorrelationCoefficient.html |access-date=2025-01-22 |website=mathworld.wolfram.com |language=en |quote=The Spearman rank correlation coefficient, also known as Spearman's rho,}}</ref>
*In [[Option (finance)|options]] theory to represent the rate of change of a portfolio with respect to [[interest rates]]<ref>{{Cite web |title=What Is Rho? Definition, How It's Used, Calculation, and Example |url=https://www.investopedia.com/terms/r/rho.asp |access-date=2025-01-22 |website=Investopedia |language=en |quote=Rho is the rate at which the price of a derivative changes relative to a change in the risk-free rate of interest. Rho measures the sensitivity of an option or options portfolio to a change in interest rate.}}</ref>

=== Chi Rho (☧) ===
[[File:Simple Labarum2.svg|thumb|181x181px|The [[Chi Rho]] symbol representing [[Jesus|Christ]].]]
The letter rho overlaid with [[chi (letter)|chi]] forms the ''[[Chi Rho]]'' symbol, used to represent [[Jesus Christ]]. It was first used by Emperor [[Constantine the Great]]. A can be seen on his standard known as the [[Labarum]].

===Rho with stroke (ϼ)===
The rho with a stroke through its tail is used for abbreviations involving rho, most notably in {{lang|el|γϼ}} for {{lang|el|γράμμα}} as a unit of measurement.<ref>[[Paul of Aegina]] explains conventional usage in the last chapter of Book VII on weights and measures and uses it throughout his work.</ref>

==Unicode==

* {{unichar|03A1|html=}}<ref>Unicode Code Charts: [https://www.unicode.org/charts/PDF/U0370.pdf Greek and Coptic (Range: 0370-03FF)]</ref>
* {{unichar|03C1|html=}}
* {{unichar|03F1|html=}}
* {{unichar|03FC}}
* {{unichar|1D29}}
* {{unichar|1D68}}
* {{unichar|1FE4}}
* {{unichar|1FE5}}
* {{unichar|1FEC}}
* {{unichar|2374}}
* {{unichar|2627}}
* {{unichar|2CA2}}
* {{unichar|2CA3}}
* {{unichar|2CE5}}
* {{unichar|2CE6}}
* {{unichar|2CE8}}
* {{unichar|2CE9}}
* {{unichar|101A0}}

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style:

* {{unichar|1D6B8}}
* {{unichar|1D6D2}}
* {{unichar|1D6E0}}
* {{unichar|1D6F2}}
* {{unichar|1D70C}}
* {{unichar|1D71A}}
* {{unichar|1D72C}}
* {{unichar|1D746}}
* {{unichar|1D754}}
* {{unichar|1D766}}
* {{unichar|1D780}}
* {{unichar|1D78E}}
* {{unichar|1D7A0}}
* {{unichar|1D7BA}}
* {{unichar|1D7C8}}

== See also ==

*[[Er (Cyrillic)|Р, р - Er (Cyrillic)]]
*[[R|R, r - Latin]]
*[[P|P, p - Latin]]

== References ==
{{Wiktionary|Ρ|ρ}}
<references/>

[[Category:Greek letters]]