Complex Numbers and Algebra - Math History - NJ Wildberger

Explore the fascinating history and applications of complex numbers in this comprehensive mathematics lecture. Delve into the origins of complex numbers, tracing their development from cubic equations to their modern geometric representation. Learn about the contributions of mathematicians like del Ferro, Tartaglia, Cardano, Euler, and Gauss. Examine the cubic equation formula and its connection to complex numbers, even when dealing with real solutions. Discover how imaginary numbers are used in calculus, including Johann Bernoulli's work on inverse tangent functions and complex logarithms. Investigate the geometric interpretation of complex numbers in the plane, from early ideas by Cotes and Euler to more explicit representations by Wessel, Argand, and Gauss. Explore the Fundamental Theorem of Algebra, its historical challenges, and the approaches taken by d'Alembert and Gauss in proving this crucial result. Gain valuable insights into the development and significance of complex numbers in mathematics.

Introduction
History of complex numbers
The cubic formula
The imaginary number
The integral
Fundamental Theorem
Geometric Representation
Argonne
Theorem of Algebra
Proof
Gauss proofs