33

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

Norbert
  • 56,803
Geek
  • 2,349
  • 3
    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 09:52
  • @AndreaMori so how does ϵ differe from δ because both are used to signify arbitrarily small quantity ? – Geek Jul 30 '12 at 10:01
  • 2
    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
  • 1
    If you don't say where you saw it, we can't give you more helpful answers... – J. M. ain't a mathematician Jul 30 '12 at 10:11
  • 6
    Not much. $ $ $ $ – Did Jul 30 '12 at 10:38
  • 8
    Paul Erdős used it to mean children as "How are the epsilons?" – Ross Millikan Jul 30 '12 at 14:03
  • 1
    see also wikipedia Epsilon(disambiguation) > Science and mathematics https://en.wikipedia.org/wiki/Epsilon_%28disambiguation%29#Science_and_mathematics , many uses for a symbol – Willemien Feb 06 '15 at 17:01

3 Answers3

54

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.

Chris Taylor
  • 28,955
  • 2
    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 09:59
  • Presumably it's $\epsilon$ for "element"..? – Chris Taylor Jul 30 '12 at 10:01
  • 3
    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
  • @sos440: not impossible? – Nikolaj-K Jul 30 '12 at 10:08
  • 1
    @NickKidman $\neg\neg p = p$ – Chris Taylor Jul 30 '12 at 10:09
  • @ChrisTaylor: Most of the times, yes. Anyway, I was thinking he wanted to say the opposite, as it's posted in response to you post. – Nikolaj-K Jul 30 '12 at 10:16
  • 5
    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
  • 1
    In formal language theory, $\varepsilon$ is sometimes used to signify the empty word. – Harald Hanche-Olsen Jul 30 '12 at 11:11
  • @ChrisTaylor Latin est can mean “it is,” but more accurately it is the third-person (i.e. not the speaker or the audience but some “third” person), singular, present-tense form of the (irregular) verb esse, “to be, to exist.” Because the form of the verb already indicates the subject is third-person singular, Latin rarely supplied a separate subject unless necessary—hence “it” is implicit in the form of the verb. But when a subject is necessary—as in $x\ ϵ\ {1, 2, 3}$—you would use the subject, x, not “it.” So “x est {1, 2, 3}” is “x is {1, 2, 3}.” – KRyan Jul 03 '22 at 17:06
5

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:

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

MJD
  • 65,394
  • 39
  • 298
  • 580
  • The Hilbert epsilon also proves the axiom of choice. If $\forall x.\exists y.\phi(x,y)$, then $\forall x.\phi(x,\varepsilon y.\phi(x,y))$. – Toby Bartels Feb 05 '20 at 18:32
3

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.

J. M. ain't a mathematician
  • 75,051