# Omega (Ω, ω) Definition

Omega (Ω, ω) is the 24th and last letter of the Greek alphabet. In the system of Greek numerals it has a value of 800. The word literally means great O (ō mega, mega meaning great), as opposed to Ο ο omicron, which means little O (o mikron, micron meaning little).

In phonetic terms, the Ancient Greek Ω is a long open-mid o [ɔː], comparable to the vowel of British English raw. In Modern Greek, Ω represents the mid back rounded vowel /o̞/, the same sound as omicron. The letter omega is transcribed ō or simply o. As the last letter of the Greek alphabet, Omega is often used to denote the last, the end, or the ultimate limit of a set, in contrast to alpha, the first letter of the Greek alphabet.

## Usage in Mathematics & Science

The uppercase Omega (Ω) is used for:

Mathematics & Computer Science:

In complex analysis, the Omega constant, a solution of Lambert's W function.

In differential geometry, the space of differential forms on a manifold (of a certain degree, usually with a superscript).

A variable for a 2-dimensional region in calculus, usually corresponding to the domain of a double integral.

In topos theory, the (codomain of the) subobject classifier of an elementary topos.

In combinatory logic, the looping combinator, (S I I (S I I))

In group theory, the omega and agemo subgroups of a p-group, Ω(G) and ℧(G)

In group theory, Cayley's Ω process as a partial differential operator.

In statistics, it is used as the symbol for the sample space, or total set of possible outcomes.

In number theory, Ω(n) is the number of prime divisors of n.

In notation related to Big O notation to describe the asymptotic behavior of functions.

Chaitin's constant.

Chemistry:

For oxygen-18, a natural, stable isotope of oxygen.

Physics:

For ohm – SI unit of electrical resistance; formerly also used upside down (℧) to represent mho, the old name for the inverse of an ohm (now siemens with symbol S) used for electrical conductance. Unicode has a separate code point for the ohm sign (U+2126, Ω), but it is included only for backward compatibility, and the Greek uppercase omega character (U+03A9, Ω) is preferred.

In statistical mechanics, Ω refers to the multiplicity (number of microstates) in a system.

The solid angle or the rate of precession in a gyroscope.

In particle physics to represent the Omega baryons.

In astronomy (cosmology), Ω refers to the density of the universe, also called the density parameter.

In astronomy (orbital mechanics), Ω refers to the longitude of the ascending node of an orbit.

As Part of a Logo or Trademark:

The logo of Omega Watches SA.

Part of the original Pioneer logo.

Part of the Badge of the Supreme Court of the United Kingdom.

Part of the mission patch for STS-135, as it was the last mission of the Space Shuttle program.

The logo of the God of War video game series based on Greek mythology. In God of War (2018), it is revealed it stands as the symbol of war in Greece.

The logo of E-123 Omega, a Sonic the Hedgehog character.

The logo of the Heroes of Olympus series, based on Greek mythology.

The logo of the Ultramarines in Warhammer 40,000.

The logo of Primal Groudon, the version mascot of Pokémon Omega Ruby.

The logo of Darkseid in DC comics.

One of the logos of professional wrestler Kenny Omega.

Other:

The symbol of the resistance movement against the Vietnam-era draft in the United States.

Year or date of death.

Used to refer to the lowest-ranked wolf in a pack.

In eschatology, the symbol for the end of everything.

In molecular biology, the symbol is used as shorthand to signify a genetic construct introduced by a two-point crossover.

Omega Particle in the Star Trek universe.

The final form of NetNavi bosses in some of the Mega Man Battle Network games.

The personal symbol for Death, as worn by Death in the Discworld series by Terry Pratchett.

The symbol to represent Groudon in Pokémon Omega Ruby and Alpha Sapphire.

A secret boss in the Final Fantasy series called Omega ( Ω ) Weapon.

A character from the series Doctor Who called Omega, believed to be one of the creators of the Time Lords of Gallifrey.

The lowercase Omega (ω) is used for:

Mathematics:

The first transfinite ordinal number, often identified with the set of natural numbers including 0 (sometimes written ω

_{0}).In set theory, the first uncountable ordinal number (more commonly written as ω

_{1}).A primitive root of unity, like the complex cube roots of 1.

The Wright Omega function.

A generic differential form.

In number theory, ω(n) is the number of distinct prime divisors of n.

In number theory, an arithmetic function.

In combinatory logic, the self-application combinator, (λ x. x x).

In mathematical/options finance, the elasticity of financial options.

In analytical investment management, the tracking error of an investment manager.

Biology, Biochemistry & Chemistry:

In biochemistry, for one of the RNA polymerase subunits.

In biochemistry, for the dihedral angle associated with the peptide group, involving the backbone atoms Cα-C'-N-Cα.

In biology, for the fitness.

In chemistry, for denoting the carbon atom furthest from the carboxyl group of a fatty acid.

In genomics, as a measure of molecular evolution in protein-coding genes (also denoted as d

_{N}/d_{S}or K_{a}/K_{s}ratio).Physics:

Angular velocity or angular frequency.

In computational fluid dynamics, the specific turbulence dissipation rate.

In meteorology, the change of pressure with respect to time of a parcel of air.

In circuit analysis and signal processing to represent natural frequency, related to frequency f by ω = 2πf.

In astronomy, as a ranking of a star's brightness within a constellation.

In orbital mechanics, as designation of the argument of periapsis of an orbit.

In particle physics to represent the omega meson.

Computer Science:

In notation related to Big O notation, the asymptotically dominant nature of functions.

In relational database theory to represent NULL, a missing or inapplicable value.

Other:

Used in place of ん in Japanese typing shorthand.

In linguistics, the phonological word.

In textual criticism, the archetype of a manuscript tradition.

In sociology, used to refer to the lowest ranking member of a group.

In shift_JIS art, used to represent the cat's mouth. (e.g. (´･λ･`) ｼｮﾎﾞｰﾝ).

## Greek Alphabet

The letters of the Ancient Greek Alphabet, which are frequently utilized in math and science:

### Related Definitions

### Sources

“Omega.” Wikipedia, Wikimedia Foundation, 30 Apr. 2020, en.wikipedia.org/wiki/Omega.