variable

variable

The word

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

variable

:

(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

,

π

,

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.

(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

function

f

,

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

some specific function

, not all functions.

(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

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:

(

∃

x

(

(

x

≤

0

)

∧

R

(

z

)

)

.

(iv) As pieces of (experimental) data.

Used in the sciences. One may say “at

t

=

4

⁢

s

,

x

=

23.1

⁢

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

d

⁢

x

/

d

⁢

t

.

If we want to talk about a specific (but unnamed) time,
we can use a notation such as “when

t

=

t

0

,

…

”
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

Ω

.
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

ω

∈

Ω

,
is denoted by

x

.

(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

←

2

”.

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.

Title

variable

Canonical

name

Variable

Date of creation

2013-03-22 15:31:39

Last modified on

2013-03-22 15:31:39

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

10

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 00A05

Related topic

Parameter

Generated on Tue Feb 6 22:18:55 2018 by

LaTeXML