Complex numbers of the form a+bi are mostly introduced these days in the context of quadratic equations, but according to Stillwell cubic equations are closer to their historical roots. We show how the cubic equation formula of del Ferro, Tartaglia and Cardano requires some understanding of complex numbers even when only real zeroes appear to be involved.
The use of imaginary numbers in calculus manipulations is illustrated with some computations of Johann Bernoulli relating the inverse tan function to complex logarithms, and the connections bewteen tan (na) to tan(a).
The geometrical planar representation of complex numbers goes back to Cotes, Euler and DeMoivre in some form, and then more explicity at the end of the 18th century to Wessel and Argand, and then Gauss.
The Fundamental theorem of algebra is a key undergraduate result that often proves elusive---it was so also for the pioneers of the subject. Euler, Gauss and d'Alembert all struggled with the result, but made progress. Here we outline the ideas behind the proofs of d'Alembert and Gauss.
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Screenshot PDFs for my videos are available at the website wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at www.researchgate.net/profile/...
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
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Here are the Insights into Mathematics Playlists:
• MathHistory: A course ...
• WildTrig: Intro to Rat...
• Math Foundations
• Wild Linear Algebra
• Famous Math Problems
• Box Arithmetic
• Elementary Mathematics...
• Year9Maths
• Ancient Mathematics
• Wild West Banking
• Sociology and Pure Mat...
• Sociology and Pure Phy...
• Old Babylonian Mathema...
• Probability and Statis...
• Boole's Logic and Circ...
• Universal Hyperbolic G...
• Differential Geometry
• Algebraic Topology
• MathSeminars
• Playing Go
• Diffusion Symmetry: A ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
• Algebraic Calculus One
• Classical to Quantum
• Algebraic Calculus Two
• Advice for mathematics...
• Solving Polynomial Equ...
• The Hexagrammum Mystic...
• Algebraic Calculus and...
• Dynamics on Graphs
• Playlist
• Triangle Geometry
• Explorations with q-se...
• Six: An elementary cou...
• Maxel Inverses and Ort...
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