Explore the logical challenges surrounding roots of unity in complex algebra through this 29-minute video lecture. Examine the rational parametrization of the unit circle and the differences between specifying unit quadrance complex numbers using rational expressions versus trigonometric functions. Investigate the difficulties in defining angles for complex numbers and the use of infinite series for circular functions. Delve into the construction of regular polygons, particularly the seven-gon, and consider purely algebraic approaches to solving equations like z^7=1 without relying on infinite processes. Gain insights into the foundational issues underlying these common algebraic objects and their geometric representations.
Rethinking Roots of Unity in Complex Algebra - Math Foundations 218
Insights into Mathematics via YouTube
Overview
Syllabus
The problem with "roots of unity"
The Unit Circle
Rational Parametrization
Quadrance and the product of unit complex numbers
Product sof two particular complex numbers
The difficulties with "angles" for complex numbers
Infinite series for circular functions
Roots of unity
How do you construct an accurate regular seven- gon?
Purely algebraic ways of solving z^7=1, without infinite processes
Taught by
Insights into Mathematics
