Editing Indeterminate (variable) (section)

{{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>
{{bsn|date=August 2024}}

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.