Complex Numbers: Introduction and Properties

This lecture introduces complex numbers as an extension of real numbers to solve equations like x² = 2 that have no solutions in the set of rational numbers. It explains the need to 'widen' the number system to include solutions to such equations, leading to the construction of the set of complex numbers. The lecture covers the definition of complex numbers, their operations, and properties, such as the modulus and the fundamental theorem of algebra. It also explores the relationship between real and complex numbers, highlighting how complex numbers simplify certain calculations and provide solutions to equations that are unsolvable in the real number system.