**Contents**:

- What is Epsilon(ε) in Calculus?
- ε-Calculus

## 1. What is Epsilon?

In calculus, **Epsilon** (ε) is a tiny number, close to zero.

You’ll come across ε in proofs, especially in the “epsilon-delta” definition of a limit. The definition gives us the limit L of a function f(x) defined on a certain interval, as x approaches some number x_{0}. For every ε > 0 there is a δ > 0 so that for every x-value:

0 <│x – x_{0}│< δ →│f(x) – L│ < ε

If you don’t quite “get” what the formula is doing, don’t be too hard on yourself because **the formula isn’t intuitive at all**. Evelyn Lamb, in her Scientific American article *The Subterfuge of Epsilon and Delta* calls the epsilon-delta proof

“…an initiation rite into the secret society of mathematical proof writers”.

In fact, while Newton and Leibniz invented calculus in the late 1600s, **it took more than 150 years to develop the rigorous δ-ε proof**s. δ-ε makes its first appearance in the works of Cauchy (Grabiner, 1983):

“Let δ, ε be two very small numbers; the first is chosen so that for all numerical…values of h less than δ, and for any value of x included [in the interval of definition], the ratio (f(x + h) = f(x) ) / h will always be greater than f′(x) – ε and less than f′ (x) + ε” (Cauchy, 1823, p. 44)

Today, εδ forms the basis of just about every proof in calculus.

## In Statistics

In regression analysis, epsilon (ε) is a measurement of how far from the true regression line the observation y is (e.g. in the equation, Y = Xβ + ε). The true regression line is the line of the means (the mean of epsilon is zero).

Uncommonly, you might also see the term epsilon-squared, which is a measure of effect size (a measure of relationship between groups).

## 2. Epsilon Calculus

ε-calculus is an extended form of predicate calculus that was developed by David Hilbert with reference to arithmetic and set theory.

ε-calculus uses an ε operator, a term-forming operator which replaces quantifiers in ordinary predicate logic (Miyamoto & Moser, 2019): For a formula A(x), εxA(x) is an ε-term.

The critical axiom in ε-calculus is TA(t) → A(εx A(x))—.

## References

Cauchy, A.L. (1823). Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal. Imprimerie royale.

Grabiner, J. (1983). Who Gave You The Epsilon? Cauchy and the Origins of Rigorous Calculus. American Mathematical Monthly 90 (1983), 185-194.

Lamb, E. (2015). The Subterfuge of ε and Delta. Retrieved September 28, 2020 from: https://blogs.scientificamerican.com/roots-of-unity/the-subterfuge-of-epsilon-and-delta/

Miyamoto, K. & Moser, G. (2019). The ε Calculus with Equality and

Herbrand Complexity. Retrieved September 28, 2020 from: http://arxiv-export-lb.library.cornell.edu/pdf/1904.11304

Slater, B. ε Calculi. Retrieved September 28, 2020 from: https://iep.utm.edu/ep-calc/

**CITE THIS AS:**

**Stephanie Glen**. "Epsilon (Calculus)" From

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