Epsilon — Ε / ε
Ε ε
| Uppercase | Ε |
|---|---|
| Lowercase | ε |
| Transliteration | e |
| Pronunciation | EHP‑see‑lon |
| Numeric value | 5 |
Etymology and Origin
Epsilon comes from the Phoenician letter "he," which represented a window or fence. The name "epsilon" means "simple e" or "bare e" (ἒ ψιλόν) to distinguish it from the diphthong "ai" which was also pronounced like "e." Uppercase epsilon (Ε) looks identical to the Latin letter E.
Pronunciation
- Ancient Greek: [e] as short "e" in "pet"
- Modern Greek: [e] as in "met" (unchanged)
- English usage: EP-sih-lon [ˈɛpsɪlɒn]
Epsilon in Calculus
Epsilon is most famous for its role in calculus as a symbol for arbitrarily small positive quantities. In the epsilon-delta definition of limits, ε represents how close the function value must be to the limit, while δ represents how close the input must be.
- Limit definition: lim(x→a) f(x) = L means: for all ε > 0, there exists δ > 0 such that |f(x) - L| < ε
- Continuity: Function continuous if for all ε > 0, exists δ > 0: |x - a| < δ implies |f(x) - f(a)| < ε
Uses of Epsilon in Mathematics
- Analysis: Arbitrarily small positive number in limits and continuity
- Error Terms: ε represents error or residual in approximations
- Set Theory: ∈ (epsilon-based) means "is an element of"
- Levi-Civita Symbol: εᵢⱼₖ in tensor calculus
- Machine Epsilon: Smallest number where 1 + ε ≠ 1 in computing
- Regex: ε-transitions in automata theory
Uses in Science and Engineering
- Physics: ε₀ (permittivity of free space) ≈ 8.854 × 10⁻¹² F/m
- Electrical Engineering: Dielectric constant (relative permittivity)
- Materials Science: Strain ε = ΔL/L (change in length/original length)
- Chemistry: Molar extinction coefficient in Beer-Lambert law
- Astronomy: Eccentricity of orbits (sometimes)
- Economics: Elasticity of demand/supply
Cultural and Symbolic Uses
- Fraternities/Sororities: Sigma Phi Epsilon, many others
- Euro Symbol: € inspired by Greek epsilon (Europe)
- Astronomy: Fifth brightest star in constellations
- Programming: Empty string often denoted by ε
Mathematical Examples with Epsilon
Limit proof: Show lim(x→2) 3x = 6. Given ε > 0, choose δ = ε/3. If |x-2| < δ, then |3x-6| = 3|x-2| < 3δ = ε ✓
Strain calculation: If a 10m beam stretches to 10.02m, ε = 0.02/10 = 0.002 = 0.2%
Permittivity: For water, εᵣ ≈ 80 (dielectric constant)
Machine epsilon: For double precision (64-bit), ε ≈ 2.22 × 10⁻¹⁶
Writing Tips
- Uppercase Ε: Identical to Latin E - three horizontal lines connected to vertical stroke
- Lowercase ε: Like a backwards "3" or a curved "c" with line through middle
- Common mistake: Don't confuse lowercase ε with "∈" (element of symbol)
- Handwriting: Lowercase ε should be written in one stroke, starting from top right
Interesting Facts
- Epsilon is the ancestor of the Latin letter E
- In ancient Greek numerals, Ε΄ = 5
- The epsilon-delta definition was formalized by Karl Weierstrass in the 1800s
- Vacuum permittivity ε₀ is one of the fundamental constants of physics
- The € (euro) symbol was officially adopted in 1996
- Epsilon Eridani is one of the closest stars to Earth (10.5 light-years)
- In formal languages, ε represents the empty string or null transition
- Google's floating-point library is called "epsilon"
Copy Epsilon
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Ε
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Unicode code points: U+0395 (uppercase), U+03B5 (lowercase).