- This article is about the zeros of a function, which should not be confused with the value at zero. You may also want information on the Nth roots of numbers instead.
In mathematics, a root (or a zero) of a complex-valued function <math>f</math> is a member <math>x</math> of the domain of <math>f</math> such that <math>f(x)</math> vanishes at <math>x</math>, that is,
- <math>x : f(x) = 0\,.</math>
In other words, a "root" of a function <math>f</math> is a value for <math>x</math> that produces a result of zero ("0"). For example, consider the function <math>f</math> defined by the following formula:
- <math>f(x)=x^2-6x+9 \,.</math>
This function has a root at 3 because <math>f(3) = 3^2 - 6(3) + 9 = 0</math>. If the function is mapping from real numbers to real numbers, its zeros are the points where its graph meets the x-axis. In this situation, a root can be called an x-intercept. The word root can also refer to a number in the form <math>a^{1/n}</math> (which is the root of the polynomial <math>x^n-a</math>) such as the square root or other roots. A substantial amount of mathematics was developed in order to find roots of various functions, especially polynomials. One wide-ranging concept, complex numbers, was developed to handle the roots of quadratic or cubic equations with negative discriminant (that is, those leading to expressions involving the square root of negative numbers). All real polynomials of odd degree have a real number as a root. Many real polynomials of even degree do not have a real root, but the fundamental theorem of algebra states that every polynomial of degree <math>n</math> has <math>n</math> complex roots, counted with their multiplicities. The non-real roots of real polynomials come in conjugate pairs. One of the most important unsolved problems in mathematics concerns the location of the roots of the Riemann zeta function.
