Rho

Rho /ˈroʊ/ (uppercase Ρ, lowercase ρ or ϱ; Greek: ῥῶ) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the same glyph, Ρ, as the distinct Latin letter P; the two letters have different Unicode encodings.

For other uses, see Rho (disambiguation).

Uses

The Greek alphabet on a black figure vessel, with an R-shaped rho.

Greek

Rho is classed as a liquid consonant (together with lambda and sometimes the nasals mu and nu), which has important implications for morphology. In both Ancient and Modern Greek, it represents a trilled or tapped r.

In polytonic orthography, a rho at the beginning of a word is written with a rough breathing, equivalent to h (ῥ rh), and a double rho within a word is written with a smooth breathing over the first rho and a rough breathing over the second (ῤῥ rrh). That apparently reflected an aspirated or voiceless pronunciation in Ancient Greek, which led to the various Greek-derived English words starting with rh or contain rrh.

The name of the letter is written in Greek as ῥῶ (polytonic) or ρω/ρο (monotonic).

Other alphabets

Letters that arose from rho include Roman R and Cyrillic Er (Р).

Mathematics and science

The characters ρ and 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⁺, ρ⁻, ρ⁰)
  • 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, cylindrical, spherical, and toroidal coordinate systems, and toroidal and poloidal coordinates of the Earth's magnetic field.
  • The correlation coefficient of a population parameter
  • The spectral radius of a matrix ${\displaystyle A}$ denoted as ${\displaystyle \rho (A)}$
  • The plastic number
  • The prime constant
  • The sensitivity to interest rates of a pricing function
  • The expected return of a given policy (${\displaystyle \pi }$) in reinforcement learning, denoted ${\displaystyle \rho ^{\pi }}$
• In economics to represent the discount rate of future pence cash flows
• In molecular biology to represent the Rho protein responsible for termination of RNA synthesis. In such occasions, it is often represented as ϱ ("rho symbol" U+03F1), 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
  • For LaTeX, the symbols are \rho (${\displaystyle \rho }$), and \varrho (${\displaystyle \varrho }$) (note: \Rho is not defined by default)
  • 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 operation in relational algebra
• In statistics to represent Spearman's rank correlation coefficient, commonly known as Spearman's rho
• In options theory to represent the rate of change of a portfolio with respect to interest rates

Chi Rho

The letter rho overlaid with chi forms the Chi Rho symbol, used to represent Jesus Christ.

Rho with stroke (ϼ)

The rho with a stroke through its tail is used for abbreviations involving rho, most notably in γϼ for γράμμα as a unit of measurement.[1]

Character encodings

Greek rho[2]

Character information

Preview	Ρ		ρ		Ῥ		ῥ		ῤ
Unicode name	GREEK CAPITAL LETTER RHO		GREEK SMALL LETTER RHO		GREEK CAPITAL LETTER RHO WITH DASIA		GREEK SMALL LETTER RHO WITH DASIA		GREEK SMALL LETTER RHO WITH PSILI
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	929	U+03A1	961	U+03C1	8172	U+1FEC	8165	U+1FE5	8164	U+1FE4
UTF-8	206 161	CE A1	207 129	CF 81	225 191 172	E1 BF AC	225 191 165	E1 BF A5	225 191 164	E1 BF A4
Numeric character reference	Ρ	Ρ	ρ	ρ	Ῥ	Ῥ	ῥ	ῥ	ῤ	ῤ
Named character reference	Ρ		ρ
DOS Greek	144	90	168	A8
DOS Greek-2	199	C7	235	EB
Windows 1253	209	D1	241	F1
TeX			\rho

Greek rho symbols

Character information

Preview	ϱ		ϼ		ᴩ		ᵨ		☧
Unicode name	GREEK RHO SYMBOL		GREEK RHO WITH STROKE SYMBOL		GREEK LETTER SMALL CAPITAL RHO		GREEK SUBSCRIPT SMALL LETTER RHO		CHI RHO
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	1009	U+03F1	1020	U+03FC	7465	U+1D29	7528	U+1D68	9767	U+2627
UTF-8	207 177	CF B1	207 188	CF BC	225 180 169	E1 B4 A9	225 181 168	E1 B5 A8	226 152 167	E2 98 A7
Numeric character reference	ϱ	ϱ	ϼ	ϼ	ᴩ	ᴩ	ᵨ	ᵨ	☧	☧
Named character reference	ϱ, ϱ
TeX	\varrho

Coptic ro

Character information

Preview	Ⲣ		ⲣ		⳩
Unicode name	COPTIC CAPITAL LETTER RO		COPTIC SMALL LETTER RO		COPTIC SYMBOL CHI RHO
Encodings	decimal	hex	decimal	hex	decimal	hex
Unicode	11426	U+2CA2	11427	U+2CA3	11497	U+2CE9
UTF-8	226 178 162	E2 B2 A2	226 178 163	E2 B2 A3	226 179 169	E2 B3 A9
Numeric character reference	Ⲣ	Ⲣ	ⲣ	ⲣ	⳩	⳩

Functional symbol

Character information

Preview	⍴
Unicode name	APL FUNCTIONAL SYMBOL RHO
Encodings	decimal	hex
Unicode	9076	U+2374
UTF-8	226 141 180	E2 8D B4
Numeric character reference	⍴	⍴

Mathematical rho

Character information

Preview	𝚸		𝛒		𝛠		𝛲		𝜌
Unicode name	MATHEMATICAL BOLD CAPITAL RHO		MATHEMATICAL BOLD SMALL RHO		MATHEMATICAL BOLD RHO SYMBOL		MATHEMATICAL ITALIC CAPITAL RHO		MATHEMATICAL ITALIC SMALL RHO
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120504	U+1D6B8	120530	U+1D6D2	120544	U+1D6E0	120562	U+1D6F2	120588	U+1D70C
UTF-8	240 157 154 184	F0 9D 9A B8	240 157 155 146	F0 9D 9B 92	240 157 155 160	F0 9D 9B A0	240 157 155 178	F0 9D 9B B2	240 157 156 140	F0 9D 9C 8C
UTF-16	55349 57016	D835 DEB8	55349 57042	D835 DED2	55349 57056	D835 DEE0	55349 57074	D835 DEF2	55349 57100	D835 DF0C
Numeric character reference	𝚸	𝚸	𝛒	𝛒	𝛠	𝛠	𝛲	𝛲	𝜌	𝜌

Character information

Preview	𝜚		𝜬		𝝆		𝝔		𝝦
Unicode name	MATHEMATICAL ITALIC RHO SYMBOL		MATHEMATICAL BOLD ITALIC CAPITAL RHO		MATHEMATICAL BOLD ITALIC SMALL RHO		MATHEMATICAL BOLD ITALIC RHO SYMBOL		MATHEMATICAL SANS-SERIF BOLD CAPITAL RHO
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120602	U+1D71A	120620	U+1D72C	120646	U+1D746	120660	U+1D754	120678	U+1D766
UTF-8	240 157 156 154	F0 9D 9C 9A	240 157 156 172	F0 9D 9C AC	240 157 157 134	F0 9D 9D 86	240 157 157 148	F0 9D 9D 94	240 157 157 166	F0 9D 9D A6
UTF-16	55349 57114	D835 DF1A	55349 57132	D835 DF2C	55349 57158	D835 DF46	55349 57172	D835 DF54	55349 57190	D835 DF66
Numeric character reference	𝜚	𝜚	𝜬	𝜬	𝝆	𝝆	𝝔	𝝔	𝝦	𝝦

Character information

Preview	𝞀		𝞎		𝞠		𝞺		𝟈
Unicode name	MATHEMATICAL SANS-SERIF BOLD SMALL RHO		MATHEMATICAL SANS-SERIF BOLD RHO SYMBOL		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL RHO		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL RHO		MATHEMATICAL SANS-SERIF BOLD ITALIC RHO SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120704	U+1D780	120718	U+1D78E	120736	U+1D7A0	120762	U+1D7BA	120776	U+1D7C8
UTF-8	240 157 158 128	F0 9D 9E 80	240 157 158 142	F0 9D 9E 8E	240 157 158 160	F0 9D 9E A0	240 157 158 186	F0 9D 9E BA	240 157 159 136	F0 9D 9F 88
UTF-16	55349 57216	D835 DF80	55349 57230	D835 DF8E	55349 57248	D835 DFA0	55349 57274	D835 DFBA	55349 57288	D835 DFC8
Numeric character reference	𝞀	𝞀	𝞎	𝞎	𝞠	𝞠	𝞺	𝞺	𝟈	𝟈

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.

See also

• Р, р - Er (Cyrillic)
• R, r - Latin
• P, p - Latin

References

1. Paul of Aegina explains conventional usage in the last chapter of Book VII on weights and measures and uses it throughout his work.
2. Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)