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{{Css Image Crop |Image = X-intercepts.svg |bSize = 300px |cWidth = 300 |cHeight = 110 |oLeft = 0 |oTop = 100 |Location = right |Description = A graph of the function cos(''x'') on the domain <math>\scriptstyle{[-2\pi,2\pi]}</math>, with ''x''-intercepts indicated in red. The function has '''zeroes''' where ''x'' is <math>\scriptstyle\frac{-3\pi}{2}</math>, <math>\scriptstyle\frac{-\pi}{2}</math>, <math>\scriptstyle\frac{\pi}{2}</math> and <math>\scriptstyle\frac{3\pi}{2}</math>.}} | |||
In [[mathematics]], a '''zero''', also sometimes called a '''root''', of a real-, complex- or generally [[vector-valued function]] ''f'' is a member ''x'' of the [[Domain of a function|domain]] of ''f'' such that ''f''(''x'') '''vanishes''' at ''x''; that is, | |||
:<math>x \text{ such that } f(x) = 0\,.</math> | |||
In other words, a "zero" of a function is an input value that produces an output of zero (0).<ref name="Foerster">{{cite book | last = Foerster | first = Paul A. | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | year = 2006 | page = 535 | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | url = http://www.amazon.com/Algebra-Trigonometry-Functions-Applications-Prentice/dp/0131657100 | isbn = 0-13-165711-9}}</ref> | |||
A '''root''' of a [[polynomial]] is a zero of the associated [[polynomial function]]. | |||
The [[fundamental theorem of algebra]] shows that any non-zero [[polynomial]] has a number of roots at most equal to its [[Degree of a polynomial|degree]] and that the number of roots and the degree are equal when one considers the [[complex number|complex]] roots (or more generally the roots in an [[algebraically closed extension]]) counted with their [[multiplicity (mathematics)|multiplicities]]. For example, the polynomial ''f'' of degree two, defined by | |||
:<math>f(x)=x^2-5x+6</math> | |||
has the two roots 2 and 3, since | |||
:<math>f(2) = 2^2 - 5 \cdot 2 + 6 = 0 \quad \textstyle{\rm {and} }\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0.</math> | |||
If the function maps [[real number]]s to real numbers, its zeroes are the ''x''-coordinates of the points where its [[Graph of a function|graph]] meets the [[x-axis|''x''-axis]]. An alternative name for such a point (''x'',0) in this context is an '''''x''-intercept'''. | |||
== Polynomial roots == | |||
{{main|Properties of polynomial roots}} | |||
Every real polynomial of odd [[Degree of a polynomial|degree]] has an odd number of real roots (counting [[Multiplicity_(mathematics)#Multiplicity_of_a_root_of_a_polynomial|multiplicities]]); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because one is the smallest odd whole number), whereas even polynomials may have none. This principle can be proven by reference to the [[intermediate value theorem]]: since polynomial functions are [[Continuous function|continuous]], the function value must cross zero in the process of changing from negative to positive or vice-versa. | |||
===Fundamental theorem of algebra=== | |||
{{main|Fundamental theorem of algebra}} | |||
The fundamental theorem of algebra states that every polynomial of degree ''n'' has ''n'' complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in [[complex conjugate|conjugate]] pairs.<ref name="Foerster" /> [[Vieta's formulas]] relate the coefficients of a polynomial to sums and products of its roots. | |||
== Computing roots == | |||
{{main|Root-finding algorithm}} | |||
{{main|Equation solving}} | |||
Computing roots of certain functions, especially [[polynomial function]]s, frequently requires the use of specialised or [[approximation]] techniques (for example, [[Newton's method]]). | |||
==Zero set== | |||
{{main|Zero set}} | |||
In topology and other areas of mathematics, the '''zero set''' of a real-valued [[function (mathematics)|function]] ''f'' : ''X'' → '''R''' (or more generally, a function taking values in some [[Abelian group|additive group]]) is the [[subset]] <math>f^{-1}(0)</math> of ''X'' (the [[inverse image]] of {0}). | |||
Zero sets are important in many areas of mathematics. One area of particular importance is algebraic geometry, where the first definition of an [[algebraic variety]] is through zero-sets. For instance, for each set ''S'' of polynomials in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], one defines the zero-locus ''Z''(''S'') to be the set of points in '''A'''<sup>''n''</sup> on which the functions in ''S'' simultaneously vanish, that is to say | |||
:<math>Z(S) = \{x \in \mathbb A^n \mid f(x) = 0 \text{ for all } f\in S\}.</math> Then a subset ''V'' of '''A'''<sup>''n''</sup> is called an '''affine algebraic set''' if ''V'' = ''Z''(''S'') for some ''S''. These affine algebraic sets are the fundamental building blocks of algebraic geometry. | |||
== See also == | |||
* [[Zero (complex analysis)]] | |||
* [[Pole (complex analysis)]] | |||
* [[Fundamental theorem of algebra]] | |||
* [[Newton's method]] | |||
* [[Sendov's conjecture]] | |||
* [[Marden's theorem]] | |||
== References == | |||
{{reflist}} | |||
==Further reading== | |||
* {{MathWorld |title=Root |urlname=Root}} | |||
[[Category:Elementary mathematics]] | |||
[[Category:Functions and mappings]] | |||
[[Category:Zero]] | |||
[[ru:Нуль функции]] | |||
Revision as of 10:20, 18 January 2014
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is,
In other words, a "zero" of a function is an input value that produces an output of zero (0).[1]
A root of a polynomial is a zero of the associated polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by
has the two roots 2 and 3, since
If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.
Polynomial roots
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because one is the smallest odd whole number), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero in the process of changing from negative to positive or vice-versa.
Fundamental theorem of algebra
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.[1] Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Computing roots
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Computing roots of certain functions, especially polynomial functions, frequently requires the use of specialised or approximation techniques (for example, Newton's method).
Zero set
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
In topology and other areas of mathematics, the zero set of a real-valued function f : X → R (or more generally, a function taking values in some additive group) is the subset of X (the inverse image of {0}).
Zero sets are important in many areas of mathematics. One area of particular importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k[x1, ..., xn], one defines the zero-locus Z(S) to be the set of points in An on which the functions in S simultaneously vanish, that is to say
- Then a subset V of An is called an affine algebraic set if V = Z(S) for some S. These affine algebraic sets are the fundamental building blocks of algebraic geometry.
See also
- Zero (complex analysis)
- Pole (complex analysis)
- Fundamental theorem of algebra
- Newton's method
- Sendov's conjecture
- Marden's theorem
References
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Further reading
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