This letter "$\varepsilon$" is called epsilon right ? What does it signify in mathematics ?
What does the letter epsilon signify in mathematics?
-
3Traditionally $\epsilon$ 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. – 2012-07-30
-
0@AndreaMori so how does ϵ differe from δ because both are used to signify arbitrarily small quantity ? – 2012-07-30
-
2First 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? – 2012-07-30
-
1If you don't say where you saw it, we can't give you more helpful answers... – 2012-07-30
-
6Not much. $ $ $ $ – 2012-07-30
-
6Paul Erdős used it to mean children as "How are the epsilons?" – 2012-07-30
-
0see also wikipedia Epsilon(disambiguation) > Science and mathematics https://en.wikipedia.org/wiki/Epsilon_%28disambiguation%29#Science_and_mathematics , many uses for a symbol – 2015-02-06
3 Answers
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.
-
2Historically, 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. – 2012-07-30
-
0Presumably it's $\epsilon$ for "element"..? – 2012-07-30
-
2A 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... – 2012-07-30
-
0@sos440: not impossible? – 2012-07-30
-
1@NickKidman $\neg\neg p = p$ – 2012-07-30
-
0@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. – 2012-07-30
-
5Actually 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". – 2012-07-30
-
1In formal language theory, $\varepsilon$ is sometimes used to signify the empty word. – 2012-07-30
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.
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:
$$\begin{eqnarray} \def\hil#1{#1(\varepsilon x. #1(x))} \exists x.\phi(x) & \equiv & \hil{\phi}\\ \forall x.\phi(x) & \equiv & \phi(\varepsilon x.\lnot\phi(x)) \end{eqnarray} $$