Indeterminate (variable)

<h2 id="polynomials">Polynomials</h2>
Main article: <a href="/facts/Polynomial/Lzak8VVx">Polynomial</a>
A polynomial in an indeterminate 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an expression of the form 
 
 
 
 
 a
 
 0
 
 
 +
 
 a
 
 1
 
 
 X
 +
 
 a
 
 2
 
 
 
 X
 
 2
 
 
 +
 …
 +
 
 a
 
 n
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n}}
 
, where the 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 are called the <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> of the polynomial. Two such polynomials are equal only if the corresponding coefficients are equal.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> In contrast, two polynomial functions in a variable 
 
 
 
 x
 
 
 {\displaystyle x}
 
 may be equal or not at a particular value of 
 
 
 
 x
 
 
 {\displaystyle x}
 
.
For example, the functions

f
        (
        x
        )
        =
        2
        +
        3
        x
        ,
        
        g
        (
        x
        )
        =
        5
        +
        2
        x
      
    
    {\displaystyle f(x)=2+3x,\quad g(x)=5+2x}

are equal when 
 
 
 
 x
 =
 3
 
 
 {\displaystyle x=3}
 
 and not equal otherwise. But the two polynomials

2
        +
        3
        X
        ,
        
        5
        +
        2
        X
      
    
    {\displaystyle 2+3X,\quad 5+2X}

are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact,

2
        +
        3
        X
        =
        a
        +
        b
        X
      
    
    {\displaystyle 2+3X=a+bX}

does not hold unless 
 
 
 
 a
 =
 2
 
 
 {\displaystyle a=2}
 
 and 
 
 
 
 b
 =
 3
 
 
 {\displaystyle b=3}
 
. This is because 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is not, and does not designate, a number.
The distinction is subtle, since a polynomial in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 can be changed to a function in 
 
 
 
 x
 
 
 {\displaystyle x}
 
 by substitution. But the distinction is important because information may be lost when this substitution is made. For example, when working in <a href="/facts/Modular_arithmetic/0wtRa5dU">modulo 2</a>, we have that:

0
        −
        
          0
          
            2
          
        
        =
        0
        ,
        
        1
        −
        
          1
          
            2
          
        
        =
        0
        ,
      
    
    {\displaystyle 0-0^{2}=0,\quad 1-1^{2}=0,}

so the polynomial function 
 
 
 
 x
 −
 
 x
 
 2
 
 
 
 
 {\displaystyle x-x^{2}}
 
 is identically equal to 0 for 
 
 
 
 x
 
 
 {\displaystyle x}
 
 having any value in the modulo-2 system. However, the polynomial 
 
 
 
 X
 −
 
 X
 
 2
 
 
 
 
 {\displaystyle X-X^{2}}
 
 is not the zero polynomial, since the coefficients, 0, 1 and −1, respectively, are not all zero.

<h2 id="formal-power-series">Formal power series</h2>
Main article: <a href="/facts/Formal_power_series/CNmaG0Vk">Formal power series</a>
A <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> in an indeterminate 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is an expression of the form 
 
 
 
 
 a
 
 0
 
 
 +
 
 a
 
 1
 
 
 X
 +
 
 a
 
 2
 
 
 
 X
 
 2
 
 
 +
 …
 
 
 {\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots }
 
, where no value is assigned to the symbol 
 
 
 
 X
 
 
 {\displaystyle X}
 
.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero. Unlike the <a href="/facts/Power_series/vYh6sTps">power series</a> encountered in calculus, questions of <a href="/facts/Convergence_(mathematics)/DTok170z">convergence</a> are irrelevant (since there is no function at play). So power series that would diverge for values of 
 
 
 
 x
 
 
 {\displaystyle x}
 
, such as 
 
 
 
 1
 +
 x
 +
 2
 
 x
 
 2
 
 
 +
 6
 
 x
 
 3
 
 
 +
 …
 +
 n
 !
 
 x
 
 n
 
 
 +
 …
 
 
 
 {\displaystyle 1+x+2x^{2}+6x^{3}+\ldots +n!x^{n}+\ldots \,}
 
, are allowed.

<h2 id="as-generators">As generators</h2>
Main article: <a href="/facts/Generator_(mathematics)/Jo1eBRbD">Generator (mathematics)</a>
Indeterminates are useful in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> for generating <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structures</a>. For example, given a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 K
 
 
 {\displaystyle K}
 
, the set of polynomials with coefficients in 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is the <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> with <a href="/facts/Polynomial_arithmetic/ukP0KfGB">polynomial addition and multiplication</a> as operations. In particular, if two indeterminates 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are used, then the polynomial ring 
 
 
 
 K
 [
 X
 ,
 Y
 ]
 
 
 {\displaystyle K[X,Y]}
 
 also uses these operations, and convention holds that 
 
 
 
 X
 Y
 =
 Y
 X
 
 
 {\displaystyle XY=YX}
 
.
Indeterminates may also be used to generate a <a href="/facts/Free_algebra/xMrT0WbT">free algebra</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
. For instance, with two indeterminates 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, the free algebra 
 
 
 
 A
 ⟨
 X
 ,
 Y
 ⟩
 
 
 {\displaystyle A\langle X,Y\rangle }
 
 includes sums of strings in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, with coefficients in 
 
 
 
 A
 
 
 {\displaystyle A}
 
, and with the understanding that 
 
 
 
 X
 Y
 
 
 {\displaystyle XY}
 
 and 
 
 
 
 Y
 X
 
 
 {\displaystyle YX}
 
 are not necessarily identical (since free algebra is by definition non-commutative).

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Indeterminate_equation/DwdC3hyK">Indeterminate equation</a></li>
<li><a href="/facts/Indeterminate_form/5Y3JWKjS">Indeterminate form</a></li>
<li><a href="/facts/Indeterminate_system/y8s187LL">Indeterminate system</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Herstein, I. N. (1975). <a href="https://archive.org/details/i-n-herstein-topics-in-algebra-2nd-edition-1975-wiley-international-editions-joh/page/n165/mode/1up?q=Indeterminate">Topics in Algebra</a>. Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 047102371X.</li>
<li>McCoy, Neal H. (1960), <a href="https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=indeterminate">Introduction To Modern Algebra</a>, Boston: <a href="/facts/Allyn_and_Bacon/RI2leS8r">Allyn and Bacon</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/68015225">68015225</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">McCoy (1960, pp. 189, 190) - McCoy, Neal H. (1960), Introduction To Modern Algebra, Boston: Allyn and Bacon, LCCN 68015225 <a href="https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=indeterminate" target="_blank">https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=indeterminate</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Joseph Miller Thomas (1974). A Primer On Roots. William Byrd Press. ASIN B0006W3EBY. <a href="https://archive.org/details/primeronroots0000jmth/mode/2up?q=indeterminate" target="_blank">https://archive.org/details/primeronroots0000jmth/mode/2up?q=indeterminate</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lewis, Donald J. (1965). Introduction to Algebra. New York: Harper & Row. p. 160. LCCN 65-15743. <a href="https://archive.org/details/introductiontoal00lewi/page/160/mode/2up?q=indeterminate" target="_blank">https://archive.org/details/introductiontoal00lewi/page/160/mode/2up?q=indeterminate</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Landin, Joseph (1989). An Introduction to Algebraic Structures. New York: Dover Publications. p. 204. ISBN 0-486-65940-2. <a href="0-486-65940-2" target="_blank">0-486-65940-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Marcus, Marvin (1978). Introduction to Modern Algebra. New York: Marcel Dekker. pp. 140–141. ISBN 0-8247-6479-X. <a href="0-8247-6479-X" target="_blank">0-8247-6479-X</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Herstein 1975, Section 3.9. - Herstein, I. N. (1975). Topics in Algebra. Wiley. ISBN 047102371X. <a href="https://archive.org/details/i-n-herstein-topics-in-algebra-2nd-edition-1975-wiley-international-editions-joh/page/n165/mode/1up?q=Indeterminate" target="_blank">https://archive.org/details/i-n-herstein-topics-in-algebra-2nd-edition-1975-wiley-international-editions-joh/page/n165/mode/1up?q=Indeterminate</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Weisstein, Eric W. "Formal Power Series". mathworld.wolfram.com. Retrieved 2019-12-02. <a href="http://mathworld.wolfram.com/FormalPowerSeries.html" target="_blank">http://mathworld.wolfram.com/FormalPowerSeries.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Indeterminate (variable) open-in-new

Indeterminate (variable)