Same lecture, three modes — pick the one that fits how you study.

1 · Algebraic Definition of ℂ0:00

The complex numbers arise by adjoining a formal symbol i to ℝ subject to the single relation i² = −1. Every element of ℂ may be written uniquely as z = a + bi with a, b ∈ ℝ; the map z ↦ a is the real part Re(z) and z ↦ b is the imaginary part Im(z).

Addition is component-wise: (a+bi)+(c+di) = (a+c)+(b+d)i. Multiplication distributes and uses i² = −1 to produce (a+bi)(c+di) = (ac−bd)+(ad+bc)i. Together these make ℂ a field — every non-zero element has a multiplicative inverse.

2 · The Argand Diagram6:14

Represent z = a + bi as the point (a, b) in the Cartesian plane. The horizontal axis carries the real part; the vertical axis the imaginary part. Under this picture, addition becomes vector addition, and modulus |z| = √(a²+b²) is Euclidean distance from the origin.

The argument arg(z) = θ is the angle the vector makes with the positive real axis, measured counter-clockwise. Polar form z = r e^{iθ}then follows directly from Euler's formula. Multiplication of two complex numbers adds their arguments and multiplies their moduli — rotation composed with scaling.

3 · Complex Conjugate and Division11:47

The conjugate of z = a+bi is z̄ = a−bi; geometrically, reflection in the real axis. The product z·z̄ = a²+b² = |z|² is always non-negative real. This is the key to division: 1/z = z̄/|z|², which maps complex division to multiplication by the conjugate.

Conjugation is an automorphism of ℂ: it respects both addition and multiplication. Polynomials with real coefficients have roots that appear in conjugate pairs — a recurring theme in applied mathematics and control theory.

Example: Divide z₁ = 3+4i by z₂ = 1+2i: multiply numerator and denominator by z̄₂ = 1−2i, giving (3+4i)(1−2i)/5 = (11−2i)/5.12:55

4 · Why Complex Differentiability Is Special18:02

A function f : ℂ → ℂ is complex-differentiable at z₀ if the limit f′(z₀) = lim_{h→0} [f(z₀+h)−f(z₀)]/h exists, where h ∈ ℂ can approach zero from any direction. This is a far stronger condition than real differentiability, because the limit must agree for every path.

Functions satisfying this condition everywhere in an open set are called holomorphic. A holomorphic function is automatically infinitely differentiable and equal to its Taylor series — a rigidity with no real analogue. This rigidity is what makes complex analysis so powerful for evaluating real integrals and solving PDEs.