# What does the letter epsilon signify in mathematics?

This letter "$\varepsilon$" is called epsilon right ? What does it signify in mathematics ?

• Traditionally $\epsilon$ is used together with $\delta$ in the definition of limit, where it denotes an arbitrarily small quantity. Else, it is just a symbol that you can attach basically to anything. – Andrea Mori Jul 30 '12 at 9:52
• First of all, I didn't say that $\delta$ is arbitrarily small. Second, if you have several independent quantities, whatevere big or small they are, you need as many symbols, don't you? – Andrea Mori Jul 30 '12 at 10:05
• If you don't say where you saw it, we can't give you more helpful answers... – J. M. is a poor mathematician Jul 30 '12 at 10:11
• Not much.   – Did Jul 30 '12 at 10:38
• Paul Erdős used it to mean children as "How are the epsilons?" – Ross Millikan Jul 30 '12 at 14:03

The greek letter epsilon, written $\epsilon$ or $\varepsilon$, is just another variable, like $x$, $n$ or $T$.

Conventionally it's used to denote a small quantity, like an error, or perhaps a term which will be taken to zero in some limit.

It's possible that you are confusing it with the set membership symbol $\in$, which is something different. When you see $x\in X$ it means that $X$ is a set, and $x$ is a member of the set. For example,

$$1\in \{1,2,3\}$$

is true, but

$$4\in\{1,2,3\}$$

is false.

• Historically, the symbol $\in$ is derived from $\epsilon$, thus it is not impossible to confuse both symbols. Also, not as ubiquitous as its primary usage, this Greek symbol $\epsilon$ or $\varepsilon$ is also used to denote the sign, including Levi-Civita symbol in physics and random sign in probability to name a few. – Sangchul Lee Jul 30 '12 at 9:59
• A little research tells me that the symbol $\epsilon$ for set membership was first used by Peano in 1889, and he said that the $\epsilon$ stood for the Latin word est, meaning "it is" or "it exists". The more you know... – Chris Taylor Jul 30 '12 at 10:05
• @NickKidman $\neg\neg p = p$ – Chris Taylor Jul 30 '12 at 10:09
• Actually in books from the 50's you can still see $\varepsilon$ being used for $\in$. This is why often you hear people talk about "epsilon relation" or "epsilon induction". – Asaf Karagila Jul 30 '12 at 10:17
• In formal language theory, $\varepsilon$ is sometimes used to signify the empty word. – Harald Hanche-Olsen Jul 30 '12 at 11:11

Hilbert's epsilon-calculus used the letter $\varepsilon$ to denote a value satisfying a predicate. If $\phi(x)$ is any property, then $\varepsilon x. \phi(x)$ is a term $t$ such that $\phi(t)$ is true, if such $t$ exists. One can define the usual existential and universal quantifiers $\exists$ and $\forall$ in terms of the $\varepsilon$ quantifier:

$$\begin{eqnarray} \def\hil#1{#1(\varepsilon x. #1(x))} \exists x.\phi(x) & \equiv & \hil{\phi}\\ \forall x.\phi(x) & \equiv & \phi(\varepsilon x.\lnot\phi(x)) \end{eqnarray}$$

Here's a not too well-known instance of the use of $\varepsilon$ in mathematics:

One somewhat well-known transformation for accelerating the convergence of a sequence is the Shanks transformation (after Daniel Shanks, who is probably more well-known for his number-theoretic contributions). What the Shanks transformation essentially does, assuming that the sequence given is a sequence of Taylor polynomials evaluated at a certain argument, is to transform this sequence of Taylor approximants into a sequence of Padé rational approximants.

The Shanks transformation of a sequence can be expressed as a ratio of two determinants, but there is a more efficient realization of this, the Wynn $\varepsilon$ algorithm:

$$\varepsilon_{k+1}^{(n)}=\varepsilon_{k-1}^{(n+1)}+\frac1{\varepsilon_{k}^{(n+1)}-\varepsilon_k^{(n)}}$$

where $\varepsilon_0^{(n)}=S_n$ is the sequence to be transformed.