Mathematical induction. Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number. From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers. Although its namesake may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction).
History[edit] None of these ancient mathematicians, however, explicitly stated the inductive hypothesis. Description[edit] Example[edit] Algebraically: Variants[edit] Vieta's formulas. In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots. Named after François Viète (more commonly referred to by the Latinised form of his name, Franciscus Vieta), the formulas are used specifically in algebra. The Laws[edit] Basic formulas[edit] Any general polynomial of degree n (with the coefficients being real or complex numbers and an ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x1, x2, ..., xn.
Equivalently stated, the (n − k)th coefficient an−k is related to a signed sum of all possible subproducts of roots, taken k-at-a-time: for k = 1, 2, ..., n (where we wrote the indices ik in increasing order to ensure each subproduct of roots is used exactly once). The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots. Generalization to rings[edit] belong to the ring of fractions of R (or in R itself if 's are computed from the 's. And. Complex number. A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation x2 = −1, that is, i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane (also called Argand plane) by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. Overview[edit] Complex numbers allow for solutions to certain equations that have no solutions in real numbers. For example, the equation . Or and. Matrix (mathematics)
Each element of a matrix is often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[4] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
The numbers, symbols or expressions in the matrix are called its entries or its elements. Matrices are commonly written in box brackets: Complex number. Matrix (mathematics) Mathematical induction. Vieta's formulas. Complex number. Matrix (mathematics) Mathematical induction. Vieta's formulas. Complex number. Matrix (mathematics) Mathematical induction.
Vieta's formulas. Complex number. Matrix (mathematics) Mathematical induction. Vieta's formulas. Complex number. Matrix (mathematics) Complex number. Matrix (mathematics) Mathematical induction. Vieta's formulas. Complex number. Matrix (mathematics) Vieta's formulas. Complex number. Matrix (mathematics) Matrix (mathematics)