**Sigma** (upper-case **Σ**, lower-case **σ**, lower-case in word-final position **ς**; Greek: σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. When used at the end of a word (when the word is not all caps), the final form (ς) is used, e.g. Ὀδυσσεύς (Odysseus); note the two sigmas in the center of the name, and the word-final sigma at the end.

## History

The shape and alphabetic position of sigma is derived from Phoenician shin 𐤔 .

### Etymology

The name of sigma, according to one hypothesis,^{[1]} may continue that of Phoenician Samekh. According to a different theory,^{[2]} its original name may have been *san* (the name today associated with another, obsolete letter), while *sigma* was a Greek innovation that simply meant "hissing", based on a nominalization of a verb σίζω (*sízō*, from earlier **sig-jō*, meaning 'I hiss').

### Lunate sigma

In handwritten Greek during the Hellenistic period (4th and 3rd centuries BC), the epigraphic form of Σ was simplified into a C-like shape.^{[3]} It is also found on coins from the fourth century BC onward.^{[4]} This became the universal standard form of sigma during late antiquity and the Middle Ages. It is today known as *lunate sigma* (upper-case Ϲ, lower-case ϲ), because of its crescent-like shape.

It is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts. The forms of the Cyrillic letter С (representing /s/) and Coptic letter Ⲥ *sima* are derived from lunate sigma.

A dotted lunate sigma (*sigma periestigmenon*, encoded at U+03FE Ͼ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line so marked is at an incorrect position. Similarly, an antisigma, or reversed sigma (Ͻ), may mark a line that is out of place. A dotted antisigma or dotted reversed sigma (*antisigma periestigmenon*: Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.

## Uses

### Greek

In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative /s/. In Modern Greek, this sound is voiced to /z/ before /m/, /n/, /v/, /ð/ or /γ/.

### African Latin alphabets

- Berber

Upper-case Σ may be used in the Berber Latin alphabet for [ʕ], though the INALCO standard uses Ɛ instead.

- Uppercase of esh

The uppercase form of sigma was re-borrowed into the Latin alphabet – more precisely, the International African Alphabet – to serve as the uppercase of modern esh (lowercase: ʃ).

### Science and mathematics

#### Upper-case

Upper-case Σ is used as a symbol for:

- the summation operator
- e.g.:

- Sigma is the addition version of factorial.
- a class of baryons in particle physics
- macroscopic cross sections in nuclear and particle physics
- self-energy in condensed matter physics
- the balance of the invoice classes and the overall amount of the debts and demands in economics
- the set of symbols that form an alphabet in linguistics and computer science
- the covariance matrix of a set of random variables in probability theory and statistics, sometimes in the form to distinguish it from the summation operator.
- The busy beaver function

#### Lower-case

Lower-case σ is used for:

- the standard deviation of a population or probability distribution in statistics
- sigma bonds in chemistry
- to represent an unknown angle in mathematics
- cross section (physics)
- velocity dispersion in astronomy
- sigma constant in organic chemistry
- the sigma receptor in biology
- a quality model for business, Six Sigma (6σ), based on the standard deviation
- sigma-algebras, sigma-fields, and sigma-finiteness in measure theory; in general terms, the symbol
*σ*serves as a shorthand for "countably", e.g. a*σ*-compact topological space is one that can be written as a countable union of compact subsets. - the generated sigma-algebra of a set is denoted
- the sum-of-divisors function in number theory
- the Stefan–Boltzmann constant
- the "sigma factor" of RNA polymerase
- a measure of electrical conductivity
- the surface charge density in electrostatics
- normal stress in continuum mechanics
- volatility of a stock generally needed for options pricing
- a syllable in phonology
- the spectrum of a matrix , denoted as , in applied mathematics
- surface tension
- the unary operation of selection on a database relation in relational algebra
- the Pauli matrices in quantum mechanics
- a target's radar cross-section (RCS) in radar jamming or electronic warfare
- the life span of a basic multicellular unit (BMU) in bone remodeling
- the damping parameter in signal processing
- a millisecond in early 20th-century physiology literature
^{[5]} - the Weierstrass sigma-function
- the Sigmoid/Logistic function in machine learning

### Groups

- Politics

During the 1930s, an upper-case Σ was in use as the symbol of the Ação Integralista Brasileira, a fascist political party in Brazil.

- Companies

Sigma Corporation uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter. Sigma Aldrich incorporate both the name and the character in their logo.

## Character encodings

- Greek Sigma

Character | Σ | σ | ς | Ϲ | ϲ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Unicode name |
GREEK CAPITAL LETTER SIGMA | GREEK SMALL LETTER SIGMA | GREEK SMALL LETTER FINAL SIGMA | GREEK CAPITAL LUNATE SIGMA SYMBOL | GREEK LUNATE SIGMA SYMBOL | |||||

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex |

Unicode | 931 | U+03A3 | 963 | U+03C3 | 962 | U+03C2 | 1017 | U+03F9 | 1010 | U+03F2 |

UTF-8 | 206 163 | CE A3 | 207 131 | CF 83 | 207 130 | CF 82 | 207 185 | CF B9 | 207 178 | CF B2 |

Numeric character reference | Σ | Σ | σ | σ | ς | ς | Ϲ | Ϲ | ϲ | ϲ |

Named character reference | Σ | σ | ς | |||||||

DOS Greek | 145 | 91 | 169 | A9 | 170 | AA | ||||

DOS Greek-2 | 207 | CF | 236 | EC | 237 | ED | ||||

Windows 1253 | 211 | D3 | 243 | F3 | 242 | F2 | ||||

TeX | \Sigma | \sigma | \varsigma |

^{[6]}

Character | Ͻ | ͻ | Ͼ | ͼ | Ͽ | ͽ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Unicode name |
GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL | GREEK SMALL REVERSED LUNATE SIGMA SYMBOL | GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL | GREEK SMALL DOTTED LUNATE SIGMA SYMBOL | GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL | GREEK SMALL REVERSED DOTTED LUNATE SIGMA SYMBOL | ||||||

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex |

Unicode | 1021 | U+03FD | 891 | U+037B | 1022 | U+03FE | 892 | U+037C | 1023 | U+03FF | 893 | U+037D |

UTF-8 | 207 189 | CF BD | 205 187 | CD BB | 207 190 | CF BE | 205 188 | CD BC | 207 191 | CF BF | 205 189 | CD BD |

Numeric character reference | Ͻ | Ͻ | ͻ | ͻ | Ͼ | Ͼ | ͼ | ͼ | Ͽ | Ͽ | ͽ | ͽ |

- Coptic Sima

Character | Ⲥ | ⲥ | ||
---|---|---|---|---|

Unicode name |
COPTIC CAPITAL LETTER SIMA | COPTIC SMALL LETTER SIMA | ||

Encodings | decimal | hex | decimal | hex |

Unicode | 11428 | U+2CA4 | 11429 | U+2CA5 |

UTF-8 | 226 178 164 | E2 B2 A4 | 226 178 165 | E2 B2 A5 |

Numeric character reference | Ⲥ | Ⲥ | ⲥ | ⲥ |

- Mathematical Sigma

Character | ∑ | 𝚺 | 𝛔 | 𝛓 | 𝛴 | 𝜎 | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Unicode name |
N-ARY SUMMATION | MATHEMATICAL BOLD CAPITAL SIGMA |
MATHEMATICAL BOLD SMALL SIGMA |
MATHEMATICAL BOLD SMALL FINAL SIGMA |
MATHEMATICAL ITALIC CAPITAL SIGMA |
MATHEMATICAL ITALIC SMALL SIGMA |
||||||

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex |

Unicode | 8721 | U+2211 | 120506 | U+1D6BA | 120532 | U+1D6D4 | 120531 | U+1D6D3 | 120564 | U+1D6F4 | 120590 | U+1D70E |

UTF-8 | 226 136 145 | E2 88 91 | 240 157 154 186 | F0 9D 9A BA | 240 157 155 148 | F0 9D 9B 94 | 240 157 155 147 | F0 9D 9B 93 | 240 157 155 180 | F0 9D 9B B4 | 240 157 156 142 | F0 9D 9C 8E |

UTF-16 | 8721 | 2211 | 55349 57018 | D835 DEBA | 55349 57044 | D835 DED4 | 55349 57043 | D835 DED3 | 55349 57076 | D835 DEF4 | 55349 57102 | D835 DF0E |

Numeric character reference | ∑ | ∑ | 𝚺 | 𝚺 | 𝛔 | 𝛔 | 𝛓 | 𝛓 | 𝛴 | 𝛴 | 𝜎 | 𝜎 |

Named character reference | ∑ |

Character | 𝜍 | 𝜮 | 𝝈 | 𝝇 | 𝝨 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Unicode name |
MATHEMATICAL ITALIC SMALL FINAL SIGMA |
MATHEMATICAL BOLD ITALIC CAPITAL SIGMA |
MATHEMATICAL BOLD ITALIC SMALL SIGMA |
MATHEMATICAL BOLD ITALIC SMALL FINAL SIGMA |
MATHEMATICAL SANS-SERIF BOLD CAPITAL SIGMA |
|||||

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex |

Unicode | 120589 | U+1D70D | 120622 | U+1D72E | 120648 | U+1D748 | 120647 | U+1D747 | 120680 | U+1D768 |

UTF-8 | 240 157 156 141 | F0 9D 9C 8D | 240 157 156 174 | F0 9D 9C AE | 240 157 157 136 | F0 9D 9D 88 | 240 157 157 135 | F0 9D 9D 87 | 240 157 157 168 | F0 9D 9D A8 |

UTF-16 | 55349 57101 | D835 DF0D | 55349 57134 | D835 DF2E | 55349 57160 | D835 DF48 | 55349 57159 | D835 DF47 | 55349 57192 | D835 DF68 |

Numeric character reference | 𝜍 | 𝜍 | 𝜮 | 𝜮 | 𝝈 | 𝝈 | 𝝇 | 𝝇 | 𝝨 | 𝝨 |

Character | 𝞂 | 𝞁 | 𝞢 | 𝞼 | 𝞻 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Unicode name |
MATHEMATICAL SANS-SERIF BOLD SMALL SIGMA |
MATHEMATICAL SANS-SERIF BOLD SMALL FINAL SIGMA |
MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL SIGMA |
MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL SIGMA |
MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL FINAL SIGMA |
|||||

Encodings | decimal | hex | decimal | hex | decimal | hex | decimal | hex | decimal | hex |

Unicode | 120706 | U+1D782 | 120705 | U+1D781 | 120738 | U+1D7A2 | 120764 | U+1D7BC | 120763 | U+1D7BB |

UTF-8 | 240 157 158 130 | F0 9D 9E 82 | 240 157 158 129 | F0 9D 9E 81 | 240 157 158 162 | F0 9D 9E A2 | 240 157 158 188 | F0 9D 9E BC | 240 157 158 187 | F0 9D 9E BB |

UTF-16 | 55349 57218 | D835 DF82 | 55349 57217 | D835 DF81 | 55349 57250 | D835 DFA2 | 55349 57276 | D835 DFBC | 55349 57275 | D835 DFBB |

Numeric character reference | 𝞂 | 𝞂 | 𝞁 | 𝞁 | 𝞢 | 𝞢 | 𝞼 | 𝞼 | 𝞻 | 𝞻 |

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

Wikimedia Commons has media related to .the letter sigma |

Look up , Σ, or σ in Wiktionary, the free dictionary.ς |

- Antisigma
- Greek letters used in mathematics, science, and engineering
- Sampi
- Stigma (letter)
- Sibilant consonant
- ∑, or summation
- "Sigm-" as a combining form, as in
*sigm*odon,*sigm*urethra, or in the derivative "sigmoid", as in sigmoid sinus, sigmoid colon, sigmoidoscopy, etc.

## References

**^**Jeffery, Lilian H. (1961).*The local scripts of archaic Greece*. Oxford: Clarendon. pp. 25–27.**^**Woodard, Roger D. (2006). "Alphabet". In Wilson, Nigel Guy.*Encyclopedia of ancient Greece*. London: Routldedge. p. 38.**^**Edward M. Thompson (1912),*Introduction to Greek and Latin paleography*, Oxford: Clarendon. p. 108, 144**^**Parthia.com:*Numismatica Font Projects*.**^**Hill, A. V. (1935). "Units and Symbols".*Nature*.**136**(3432): 222. doi:10.1038/136222a0.**^**Unicode Code Charts: Greek and Coptic (Range: 0370-03FF)

Vijayan Ettickan