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Epsilon (Ε, ε) Definition

Epsilon (Ε, ε) or lunate ϵ or Greek: έψιλον, is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel /e/. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He. Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.

The name of the letter was originally εἶ (Ancient Greek: [êː]), but the name was changed to ἒ ψιλόν (e psilon "simple e") in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.

Usage in Mathematics & Science

The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the Latin letter E. However, it is commonly used in structural mechanics with Young's Modulus equations for calculating tensile, compressive and areal strain.

The Greek lowercase epsilon ε, the lunate epsilon symbol ε, or the Latin lowercase epsilon ɛ (see above) is used in a variety of places:

  • In engineering mechanics, strain calculations ε = increase of length / original length. Usually this relates to extensometer testing of metallic materials.

  • In mathematics:

    • (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.

      • In reference to this, the late mathematician Paul Erdős also used the term epsilons to refer to children.

    • Hilbert introduced epsilon terms εχ. ∅ as an extension to first order logic; see epsilon calculus.

    • It is used to represent the Levi-Civita symbol.

    • It is used to represent dual numbers: a + bε, with ε2 = 0 and ε ≠ 0.

    • It is sometimes used to denote the Heaviside step function.

    • In set theory, the epsilon numbers are ordinal numbers that satisfy the fixed point ε = ωε. The first epsilon number, ε0, is the limit ordinal of the set {ω, ωω, ωωω, ...}.

  • In computer science, it often represents the empty string, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter lambda (λ).

  • In computer science, the machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.

  • In physics:

    • It indicates the permittivity of a medium; with the subscript 0 (ε0) it is the permittivity of free space.

    • It can also indicate the strain of a material (a ratio of extensions).

  • In automata theory, it shows a transition that involves no shifting of an input symbol.

  • In astronomy:

    • It stands for the fifth-brightest star in a constellation (see Bayer designation).

    • Epsilon is the name for the most distant and most visible ring of Uranus.

    • In planetary science, ε denotes the axial tilt.

  • In chemistry, it represents the molar extinction coefficient of a chromophore.

  • In economics, ε refers to elasticity.

  • In statistics:

    • It is used to refer to error terms.

    • It also can to refer to the degree of sphericity in repeated measures ANOVAs.

  • In agronomy, it is used to represent the "photosynthetic efficiency" of a particular plant or crop.

Greek Alphabet

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

Greek Alphabet

Symbol

Letter

Symbol

Letter

Uppercase

Lowercase

Uppercase

Lowercase

Α

α

Alpha

Ν

ν

Nu

Β

β

Beta

Ξ

ξ

Xi

Γ

γ

Gamma

Ο

ο

Omicron

Δ

δ

Delta

Π

π

Pi

Ε

ε

Epsilon

Ρ

ρ

Rho

Ζ

ζ

Zeta

Σ

σ

Sigma

Η

η

Eta

Τ

τ

Tau

Θ

θ

Theta

Υ

υ

Upsilon

Ι

ι

Iota

Φ

φ

Phi

Κ

κ

Kappa

Χ

χ

Chi

Λ

λ

Lambda

Ψ

ψ

Psi

Μ

μ

Mu

Ω

ω

Omega

Related Definitions

Sources

Epsilon. 3 May 2020, en.wikipedia.org/wiki/Epsilon.

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