Elementary Abstract Algebra- Examples and Applications, 2019a

CONTENTS 5
2 Complex Numbers 26
2.1 The origin of complex numbers . . . . . . . . . . . . . . 26
2.1.1 A number that can’t be real (and we can prove it!) . . 26
2.1.2 Unreal, but unavoidable . . . . . . . . . . . . . . . . . 30
2.1.3 A mathematical revolution . . . . . . . . . . . . . . . 31
2.2 Arithmetic with complex numbers . . . . . . . . . . . . . 36
2.2.1 Complex arithmetic . . . . . . . . . . . . . . . . . . . 36
2.2.2 Comparison of integer, rational, real and complex addition
properties . . . . . . . . . . . . . . . . . . . . . 41
2.2.3 Comparison of integer, rational, real and complex multiplication
properties . . . . . . . . . . . . . . . . . . . 42
2.2.4 Modulus and complex conjugate . . . . . . . . . . . . 43
2.3 Alternative representations of complex numbers . . . . . 47
2.3.1 Cartesian representation of complex numbers . . . . . 47
2.3.2 Vector representation of complex numbers . . . . . . . 48
2.3.3 Polar representation of complex numbers . . . . . . . 48
2.3.4 Converting between rectangular and polar form . . . . 49
2.3.5 Multiplication and powers in complex polar form . . . 53
2.3.6 A Remark on representations of complex numbers . . 60
2.4 Complex numbers and roots of algebraic equations . . . . . . 61
2.4.1 Roots of unity and regular polygons . . . . . . . . 61
2.4.2 Complex nth roots in general . . . . . . . . . . . 68
2.4.3 Complex roots of polynomial equations . . . . . . 71
2.5 Applications of complex numbers . . . . . . . . . . . . . 75
2.5.1 General remarks on the usefulness of complex numbers 75
2.5.2 Complex numbers in electrical engineering: phasors . 75
2.5.3 Complex numbers and fractals: the Mandelbrot set . . 81
2.6 Hints for “Complex Numbers” exercises . . . . . . . . . . . . 84
2.7 Study guide for “Complex Numbers” chapter . . . . . . . . . 86