variable Variable 2005-10-05 21:48:51 2007-12-22 15:39:27 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %\usepackage{amsthm} \usepackage{enumerate} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % making logically defined graphics %\usepackage{xypic} % define commands here \newcommand{\complex}{\mathbb{C}} \newcommand{\real}{\mathbb{R}} \newcommand{\rat}{\mathbb{Q}} \newcommand{\nat}{\mathbb{N}} \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{\absW}[1]{\left\lvert#1\right\rvert} \providecommand{\absB}[1]{\Bigl\lvert#1\Bigr\rvert} \providecommand{\norm}[1]{\lVert#1\rVert} \providecommand{\normW}[1]{\left\lVert#1\right\rVert} \providecommand{\normB}[1]{\Bigl\lVert#1\Bigr\rVert} \providecommand{\defnterm}[1]{\emph{#1}} \DeclareMathOperator{\D}{D} \DeclareMathOperator{\linspan}{span} The word \emph{variable} as used in mathematics (and in other scientific fields that use mathematics) is somewhat vague and may have different meanings depending on the context. Variables are usually denoted by a single Roman or Greek letter, e.g. $x$, although sometimes a whole word or phrase can be used also. Here is a list of some of the meanings of \emph{variable}: \begin{description} \item[(i) As ``mathematical'' variables.] These stand for a concrete object, for example, an element of the real numbers That is, when we write the symbol $x$, it is a stand-in for various numbers: e.g. $2, 3, \pi, e, 578.24$. But we do not name these numbers specifically, because we may want to talk about all these numbers at once, in a general statement, theorem, or proof about numbers. Sense (i) is probably the most common usage in mainstream mathematics. \item[(ii) As placeholders in functional notation.] For example, we may be defining a function using the phrase ``define the function $f(z) = z^2 + 4$ for complex numbers $z$. This usage of a variable is slightly different from sense (i), because our objective is to talk about the \emph{function} $f$, \emph{and not its value} at a number $z$ which is $f(z)$. The notation ``$f(z) = z^2 + 4$'' is merely a much more convenient way of saying: ``define the function $f$ which takes a complex number, multiplies it by itself, and then adds four to it''. It could also be rephrased this way: ``define a function $f$ such that the statement $f(z) = z^2 + 4$ is true for all complex numbers $z$ (in sense (i))''. On the other hand, the symbol $f$, if we were to contemplate it as a ``variable'', arguably belongs to the sense (i); in this case we are talking about \emph{some specific function}, not all functions. \item[(iii) As ``formal'' variables.] For instance, we may talk about a formal polynomial $p(x) = 1 + x + x^2$. This is similar to sense (ii), but is not exactly the same. The variable $x$ here is not necessarily a complex number, or in any fixed domain at all. It is a formal symbol, which we later replace by actual elements of the real numbers, or matrices, etc. at our whim. And $p$ here is \emph{not a function}; it is a polynomial. The variables used in formal logic can also be considered to fall in sense (iii). For example, we may have a set of variables $ \{ x, y, z \}$ and a formula from the first-order language using such variables: $( \exists x (( x \leq 0) \land \mathrm{R}(z) )$. \item[(iv) As pieces of (experimental) data.] Used in the sciences. One may say ``at $t=4 \: \mathrm{s}$, $x=23.1 \: \mathrm{m}$'' which may really mean: ``at 4 seconds from the start of the experiment, the object is 23.1 metres to the right of its initial position''. So the symbols $t$ and $x$ are being used in the meaning of ``time'' and ``position'' in general. There may or may not be a functional relation between the ``variables'' $t$ and $x$. If there is, we might say ``$x$ is a function of $t$'', and we can talk about quantities such as $dx/dt$. If we want to talk about a specific (but unnamed) time, we can use a notation such as ``when $t = t_0, \dotsc$'' for some variable $t_0$ in sense (i). The field of probability and statistics follows a similar practice for what are termed ``random variables'', which are really functions defined on a measure space $\Omega$. But in practice they are usually denoted with variable notation: e.g. ``the random variable $X$'', and a specific value of this random variable $X$, at some unspecified $\omega \in \Omega$, is denoted by $x$. \item[(v) As state variables in computer algorithms.] In this case, a variable $x$ stands for a computer memory location. Or in more abstract language, $x$ is a name for a container which may hold some object. The contents of this container may change as time passes or when it is modified by a program that the computer is executing. In formal language, putting a value in the container is often denoted by notation like ``$x \leftarrow 2$''. \end{description} Note that the above distinctions are not always clear-cut. and the same symbol $x$ may be used for different purposes at once, which of course, may lead to confusion.