Numerical Analysis Class 1: Number Systems, Solving Polynomial Equations, Intermediate Value Theorem

Описание: What are rational numbers? Irrational numbers? Real numbers? Complex numbers? Algebraic numbers? Transcendental numbers? How are polynomial equations solved: linear, quadratic, cubic, etc...? Synthetic division. Numerical approximation of solutions, including on Mathematica. Intermediate Value Theorem and proof of the existence of a solution to cos x = x. Burden, Faires, Burden "Numerical Analysis": https://amzn.to/3DBvCWn

Numerical Analysis, Class 1

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(0:00) What is a rational number?
(2:28) What is an irrational number?
(3:20) Real vs complex numbers
(5:49) Algebraic vs transcendental numbers
(6:29) What is the nature of √2?
(7:31) What is the nature of π?
(9:28) Venn diagram of number system set inclusions
(11:17) Solution of a linear equation
(13:23) Example linear equation solution
(14:23) Solutions of quadratic equations (quadratic formula)
(16:08) Example quadratic equation solution
(18:22) Solutions of cubic equations (use Mathematica)
(20:48) Cubic example (use synthetic division after guessing roots from a graphing calculator)
(25:45) Rational Root Theorem comments
(26:21) Fundamental Theorem of Algebra comments
(28:13) Solutions of quaratic equations (use Mathematica)
(28:53) Quintic equations (Galois and Abel)
(31:02) Numerical solutions (numerical approximations of true exact solutions)
(32:49) TI Calculator numerical solution of a cubic
(34:08) Mathematica FindRoot, Solve, NSolve
(37:16) FindRoot to solve cos x = x on Mathematica
(38:14) Intermediate Value Theorem (IVT)
(42:23) Prove cos x = x has a solution (existence of a solution) with the Intermediate Value Theorem


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