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Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Thanks to viewer "David O" for this correction:
The claim that Euler didn’t know what to do with negative numbers, or thought they were greater than infinity, is a misinterpretation of his On Divergent Series paper. Euler argued that, for infinite series, the word “sum” should mean the original finite expression from which the series originates. For example, applying the geometric series formula "1/(1–r) = 1 + r + r^2 + r^3 + …" with r = 2 gives the formal result "–1 = 1 + 2 + 4 + 8 + …”. However, he did not mean that negative numbers themselves are infinite, nor that such “sums” are equal in the ordinary arithmetic sense for divergent series.
Source: Paragraphs 1-12 of https://arxiv.org/pdf/1808.02841
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: http://www.welchlabs.com/resources.
