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Mu (upper case Μ, lower case μ) is the 12th letter of the Greek alphabet. In the system of Greek numerals it has a value of 40.

Mu was derived from the Egyptian hieroglyphic symbol for water which had been simplified by the Phoenicians and named after their word for water, mem. The sound "M" is still connected with the sea in many languages, as in Hebrew marah, Latin mare, German Meer, and French mer.

The letter mu appears in conjunction with alpha and omega to signify the "beginning, middle (meson) and end," a phrase first found in an Orphic verse describing Zeus, and later adopted to describe both Jehovah and Jesus.

In Aeschylus' Eumenides, the repeated moaning of the letter mu is the sound made by the sleeping Furies as the ghost of Clytemnestra begins to invoke them. It again appears as an ominous mantra in a 10th century Coptic papyrus, containing a Christian curse against perjurers that invokes the angel Temeluchos:

I adjure you by the seven perfect letters, ΜΜΜΜΜΜΜ. You must appear to him, you must appear to him. I adjure you by the seven angels around the throne of the father.

The lower-case letter mu is used as a special symbol in many academic fields. The upper case Mu isn't generally used in this way because it is normally indistinguishable from the latin M.

In mathematics:

the Möbius function in number theory
the population mean or expected value in probability and statistics
a measure in measure theory

In measurement:

the SI prefix micro-, which represents one millionth, or 10−6. Unicode actually encodes the micro sign separately from the greek letter though they look identical in most fonts.
the micron, an old unit which corresponds to the micrometre (which is now denoted "µm")

In classical physics and engineering:

the coefficient of friction
reduced mass in the two-body problem
permeability in electromagnetism

In particle physics:

the elementary particle called the muon

Rarely, the letter μ may be used to distinguish one item represented by "m" from another (but a better option might be to use "m", "m", "m", etc.).

Letters that arose from the Greek Μ include the Latin M and Cyrillic М.

In computing

In unicode the upper and lower case Mu are encoded at U+039C and U+3BC respectively. In ISO 8859-7 they are encoded at CCHEX and ECHEX. The micro sign is considered a distinct character by unicode for historical reasons and is found at U+00B5 as well as position B5HEX in ISO 8859-1, 3, 8, 9, 13 and 15. ISO-8859-5 also has a character that looks somewhat like lower case Mu at E6HEX but this is actually supposed to be the "cyrillic small letter tse".

References

Moralia, by Plutarch

Ancient Christian Magic, by M. Meyer and R. Smith (ed.), Princeton University Press, ISBN 0691004587

Greek alphabet
Α α Alpha	Β β Beta	Γ γ Gamma
Δ δ Delta	Ε ε Epsilon	Ζ ζ Zeta
Η η Eta	Θ θ Theta	Ι ι Iota
Κ κ Kappa	Λ λ Lambda	Μ μ Mu
Ν ν Nu	Ξ ξ Xi	Ο ο Omicron
Π π Pi	Ρ ρ Rho	Σ σ Sigma
Τ τ Tau	Υ υ Upsilon	Φ φ Phi
Χ χ Chi	Ψ ψ Psi	Ω ω Omega

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