root

(noun)

A zero (of a function).

Related Terms

Examples of root in the following topics:

  • Newton's Method

    • Newton's Method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • Given a function ƒ defined over the reals x, and its derivative ƒ ', we begin with a first guess x0 for a root of the function f.
    • This $x$-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
    • We see that $x_{n+1}$ is a better approximation than $x_n$ for the root $x$ of the function $f$.

  • Absolute Convergence and Ratio and Root Tests

    • The root test is a criterion for the convergence (a convergence test) of an infinite series.
    • The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test, or Cauchy's radical test.
    • For a series $\sum_{n=1}^\infty a_n$, the root test uses the number $C = \limsup_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$, where "lim sup" denotes the limit superior, possibly ∞.
    • Note that if $\lim_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$ converges, then it equals $C$ and may be used in the root test instead.
    • The root test states that

  • Inverse Functions

    • Notice that neither the square root nor the principal square root function is the inverse of $x^2$ because the first is not single-valued, and the second returns $-x$ when $x$ is negative.

  • Tips for Testing Series

    • Root test: For $r = \limsup_{n \to \infty}\sqrt[n]{ \left|a_n \right|}$, if $r < 1$, then the series converges; if $r > 1$, then the series diverges; if $r = 1$, the root test is inconclusive.

  • Linear and Quadratic Functions

    • The solutions to the equation are called the roots of the equation.

  • Further Transcendental Functions

    • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.

  • Logarithmic Functions

    • For the definition to work, it must be understood that ' raising two to the 0.3219 power' means 'raising the 10000th root of 2 to the 3219th power'.
    • The tenthousandth root of 2 is 1.0000693171 and this number raised to the 3219th power is 1.2500, therefore ' 2 multiplied by itself 3.3219 times' will be 2 x 2 x 2 x 1.2500 namely 10.

  • Inverse Functions

    • To undo use the square root operation.

  • Trigonometric Substitution

    • One may use the trigonometric identities to simplify certain integrals containing radical expressions (or expressions containing $n$th roots).

  • Real Numbers, Functions, and Graphs

    • The real numbers include all the rational numbers, such as the integer -5 and the fraction $\displaystyle \frac{4}{3}$, and all the irrational numbers such as $\sqrt{2}$ (1.41421356… the square root of two, an irrational algebraic number) and $\pi$ (3.14159265…, a transcendental number).