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Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal. The smallest epsilon number ε 0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε 0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem).
In TeX, \epsilon ( ) denotes the lunate form, while \varepsilon ( ) denotes the epsilon number. Unicode versions 2.0.0 and onwards use ɛ as the lowercase Greek epsilon letter, [5] but in version 1.0.0, ϵ was used. [6] The lunate or uncial epsilon provided inspiration for the euro sign, €.
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epsilon 1. An epsilon number is an ordinal α such that α=ω α 2. Epsilon zero (ε 0) is the smallest epsilon number equinumerous Having the same cardinal number or number of elements, used to describe two sets that can be put into a one-to-one correspondence. equipollent Synonym of equinumerous equivalence class
This alternative definition is significantly more widespread: machine epsilon is the difference between 1 and the next larger floating point number.This definition is used in language constants in Ada, C, C++, Fortran, MATLAB, Mathematica, Octave, Pascal, Python and Rust etc., and defined in textbooks like «Numerical Recipes» by Press et al.
E0 or E00 can refer to: . ε 0, in mathematics, the smallest member of the epsilon numbers, a type of ordinal number; ε 0, in physics, vacuum permittivity, the absolute dielectric permittivity of classical vacuum
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε , where a and b are real numbers , and ε is a symbol taken to satisfy ε 2 = 0 {\displaystyle \varepsilon ^{2}=0} with ε ≠ 0 {\displaystyle \varepsilon \neq 0} .
2. Denotes that a number is positive and is read as plus. Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2. 3. Sometimes used instead of for a disjoint union of sets. − 1.