The roots (sometimes also called "zeros") of an equation
are the values of equation is satisfied.
Roots sets
are usually preceded by a modifier to indicate such, e.g., rational root ,
real
root , and complex root.
The fundamental theorem of algebra states that every polynomial equation of degree complex roots, counted with
multiplicity . Some roots may have multiplicity
greater than 1 (in which case they are said to be degenerate). In the Wolfram
Language , the expression Root f ,
k ] represents the polynomial equation function representing
a polynomial . Here, Wolfram
Language 's ordering.
The similar concept of the "complex
number n th
root
The roots of a complex function level
sets of its real part and imaginary
part . For an analytic function , its real
part and imaginary part satisfy the Cauchy-Riemann
equations . For example, the plot above shows the curves representing the real
part and imaginary part of points .
Householder (1970) gives an algorithm for constructing root-finding algorithms with an arbitrary order
of convergence . Special root-finding techniques can often be applied when the
function in question is a polynomial .
See also 13th Root ,
Airy Function Zeros ,
Bessel Function Zeros ,
Descartes' Sign Rule ,
Fundamental
Theorem of Symmetric Functions ,
Inside-Outside
Theorem ,
Isograph ,
Multiplicity ,
n th Root,
Polynomial ,
Polynomial Roots ,
Real-Rooted
Polynomial ,
Root Bracketing ,
Root
Dragging Theorem ,
Root Extraction ,
Root
Separation ,
Rouché's Theorem ,
Simple
Root ,
Square Root ,
Sturm
Function ,
Sturm Theorem ,
Vanishing ,
Zero ,
Zero Set Explore this topic in the MathWorld classroom
Explore with Wolfram|Alpha
References Arfken, G. "Appendix 1: Real Zeros of a Function." Mathematical
Methods for Physicists, 3rd ed. Boyer, C. B. A
History of Mathematics. Householder, A. S.
The
Numerical Treatment of a Single Nonlinear Equation. Kravanja, P. and van Barel, M. Computing
the Zeros of Analytic Functions. McNamee,
J. M. "A Bibliography on Roots of Polynomials." J. Comput. Appl.
Math. 47 , 391-394, 1993. https://doi.org/10.1016/0377-0427(93)90064-I . Press,
W. H.; Teukolsky, S. A.; Vetterling, W. T.; and Flannery, B. P.
"Roots of Polynomials." §9.5 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Whittaker, E. T.
and Robinson, G. "The Numerical Solution of Algebraic and Transcendental Equations."
Ch. 6 in The
Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. Referenced on Wolfram|Alpha Root
Cite this as:
Weisstein, Eric W. "Root." From MathWorld https://mathworld.wolfram.com/Root.html
Subject classifications More... Less...
for which the equation is satisfied.
which belong to certain sets
are usually preceded by a modifier to indicate such, e.g., 
is called a rational root,
is called a real
root, and 
is called a complex root.
has exactly 
complex roots, counted with
multiplicity. Some roots may have multiplicity
greater than 1 (in which case they are said to be degenerate). In the Wolfram
Language, the expression Root[f,
k] represents the 
th
root of the polynomial equation 
, where 
is a pure function representing
a polynomial. Here, 
, ..., 
is an index indicating the root number in the Wolfram
Language's ordering.
th
root" 
of a complex
number 
is known as an nth
root.
are the intersections of the zero-level
sets of its real part and imaginary
part. For an analytic function, its real
part and imaginary part satisfy the Cauchy-Riemann
equations. For example, the plot above shows the curves representing the real
part and imaginary part of 
, with the three roots indicated as black points.
