Editing Indeterminate (variable)

{{Short description|Symbol treated as a mathematical variable}}
{{More citations needed|date=December 2019}}

In [[mathematics]], an '''indeterminate''' or '''formal variable''' is a [[Variable (mathematics)|variable]] (a [[mathematical symbol|symbol]], usually a letter) that is used purely formally in a [[mathematical expression]], but does not stand for any value.<ref>{{harvtxt|McCoy|1960|pp=189,190}}</ref><ref>{{Cite book |last=Joseph Miller Thomas |url=https://archive.org/details/primeronroots0000jmth/mode/2up?q=indeterminate |title=A Primer On Roots |date=1974 |publisher=William Byrd Press |asin=B0006W3EBY}}</ref>
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In [[mathematical analysis|analysis]], a mathematical expression such as {{tmath|3x^2 + 4x}} is usually taken to represent a quantity whose value is a [[function (mathematics)|function]] of its variable {{tmath|x}}, and the variable itself is taken to represent an unknown or changing quantity. Two such functional expressions are considered equal whenever their value is equal for every possible value of {{tmath|x}} within the [[Domain of a function|domain of the functions]]. In [[abstract algebra|algebra]], however, expressions of this kind are typically taken to represent [[mathematical object|objects]] in themselves, elements of some [[algebraic structure]] – here a [[polynomial]], element of a [[polynomial ring]]. A polynomial can be formally defined as the [[sequence]] of its [[coefficients]], in this case {{tmath|[0, 4, 3]}}, and the expression {{tmath|1= 3x^2 + 4x}} or more explicitly {{tmath|0 x^0 + 4x^1 + 3x^2}} is just a convenient alternative notation, with powers of the indeterminate {{tmath|x}} used to indicate the order of the coefficients. Two such formal polynomials are considered equal whenever their coefficients are the same. Sometimes these two concepts of equality disagree.

Some authors reserve the word ''variable'' to mean an unknown or changing quantity, and strictly distinguish the concepts of ''variable'' and ''indeterminate''. Other authors indiscriminately use the name ''variable'' for both.

Indeterminates occur in [[polynomial]]s, [[rational fraction]]s (ratios of polynomials), [[formal power series]], and, more generally, in [[expression (mathematics)|expression]]s that are viewed as independent objects.

A fundamental property of an indeterminate is that it can be substituted with any mathematical expressions to which the same [[operation (mathematics)|operation]]s apply as the operations applied to the indeterminate.

Some authors of [[abstract algebra]] textbooks define an ''indeterminate'' over a [[ring (mathematics)|ring]] {{mvar|R}} as an element of a larger ring that is [[transcendental element|transcendental]] over {{mvar|R}}.<ref>{{Cite book |last=Lewis |first=Donald J. |url=https://archive.org/details/introductiontoal00lewi/page/160/mode/2up?q=indeterminate |title=Introduction to Algebra |publisher=[[Harper & Row]] |year=1965 |location=New York |lccn=65-15743|page=160}}</ref><ref>{{Cite book |last=Landin |first=Joseph |url=https://archive.org/details/introductiontoal00land/page/204/mode/2up?q=indeterminate |title=An Introduction to Algebraic Structures |publisher=[[Dover Publications]] |year=1989 |isbn=0-486-65940-2 |location=New York|page=204}}</ref><ref>{{Cite book |last=Marcus |first=Marvin |url=https://archive.org/details/introductiontomo0000marc/page/138/mode/2up?q=indeterminate |title=Introduction to Modern Algebra |publisher=[[Marcel Dekker]] |year=1978 |isbn=0-8247-6479-X |location=New York|pages= 140–141}}</ref> This uncommon definition implies that every [[transcendental number]] and every nonconstant polynomial must be considered as indeterminates.

==Polynomials==
{{main|Polynomial}}
A polynomial in an indeterminate <math>X</math> is an expression of the form <math>a_0 + a_1X + a_2X^2 + \ldots + a_nX^n</math>, where the ''<math>a_i</math>'' are called the [[coefficient]]s of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.<ref>{{harvnb|Herstein|1975|loc=Section 3.9}}.</ref> In contrast, two polynomial functions in a variable ''<math>x</math>'' may be equal or not at a particular value of ''<math>x</math>''.

For example, the functions
:<math>f(x) = 2 + 3x, \quad g(x) = 5 + 2x</math>

are equal when ''<math>x = 3</math>'' and not equal otherwise. But the two polynomials

:<math>2 + 3X, \quad 5 + 2X</math>
are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,
:<math>2 + 3X = a + bX</math>

does not hold ''unless'' ''<math>a = 2</math>'' and ''<math>b = 3</math>''. This is because ''<math>X</math>'' is not, and does not designate, a number.

The distinction is subtle, since a polynomial in ''<math>X</math>'' can be changed to a function in ''<math>x</math>'' by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in [[Modular arithmetic|modulo 2]], we have that:

:<math>0 - 0^2 = 0, \quad 1 - 1^2 = 0,</math>

so the polynomial function ''<math>x - x^2</math>'' is identically equal to 0 for ''<math>x</math>'' having any value in the modulo-2 system. However, the polynomial ''<math>X - X^2</math>'' is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.

==Formal power series==
{{main|Formal power series}}
A [[formal power series]] in an indeterminate ''<math>X</math>'' is an expression of the form <math>a_0 + a_1X + a_2X^2 + \ldots</math>, where no value is assigned to the symbol ''<math>X</math>''.<ref>{{Cite web|url=http://mathworld.wolfram.com/FormalPowerSeries.html|title=Formal Power Series|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero. Unlike the [[power series]] encountered in calculus, questions of [[Convergence (mathematics)|convergence]] are irrelevant (since there is no function at play). So power series that would diverge for values of ''<math>x</math>'', such as ''<math>1 + x + 2x^2 + 6x^3 + \ldots + n!x^n + \ldots\,</math>'', are allowed.

==As generators==
{{main|Generator (mathematics)}}
Indeterminates are useful in [[abstract algebra]] for generating [[mathematical structure]]s. For example, given a [[field (mathematics)|field]] ''<math>K</math>'', the set of polynomials with coefficients in ''<math>K</math>'' is the [[polynomial ring]] with [[polynomial arithmetic|polynomial addition and multiplication]] as operations. In particular, if two indeterminates ''<math>X</math>'' and ''<math>Y</math>'' are used, then the polynomial ring ''<math>K[X,Y]</math>'' also uses these operations, and convention holds that ''<math>XY=YX</math>''.

Indeterminates may also be used to generate a [[free algebra]] over a [[commutative ring]] ''<math>A</math>''. For instance, with two indeterminates ''<math>X</math>'' and ''<math>Y</math>'', the free algebra ''<math>A\langle X,Y \rangle</math>'' includes sums of strings in ''<math>X</math>'' and ''<math>Y</math>'', with coefficients in ''<math>A</math>'', and with the understanding that ''<math>XY</math>'' and ''<math>YX</math>'' are not necessarily identical (since free algebra is by definition non-commutative).

==See also==
*[[Indeterminate equation]]
*[[Indeterminate form]]
*[[Indeterminate system]]

==Notes==
{{reflist}}

==References==
*{{cite book |isbn=047102371X|title=Topics in Algebra|last1=Herstein|first1=I. N.|year=1975|publisher=Wiley |url = https://archive.org/details/i-n-herstein-topics-in-algebra-2nd-edition-1975-wiley-international-editions-joh/page/n165/mode/1up?q=Indeterminate}}
* {{ citation | last1 = McCoy | first1 = Neal H. | title = Introduction To Modern Algebra | location = Boston | publisher = [[Allyn and Bacon]] | year = 1960 | lccn = 68015225 | url = https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=indeterminate }}

[[Category:Abstract algebra]]
[[Category:Polynomials]]
[[Category:Series (mathematics)]]