Modulus & Argument

The conjugate reflects z across the real axis in the Argand plane. Re(z̄) = Re(z) and Im(z̄) = −Im(z). Geometrically: same distance from origin, opposite angle.

Four fundamental identities. z·z̄ = a²+b² = |z|² is the most used — it converts division into multiplication. Also: conjugate distributes over addition and multiplication: z₁+z₂ bar = z̄₁+z̄₂, and z₁z₂ bar = z̄₁·z̄₂.

|z| is the distance of z from the origin in the Argand plane. Always a non-negative real number. |z| = 0 iff z = 0. Follows directly from the Pythagorean theorem on the right triangle formed by a, b, and |z|.

Modulus is multiplicative — it converts products into products of real numbers. Also: |Re(z)| ≤ |z| and |Im(z)| ≤ |z|, since each component cannot exceed the full distance from origin.

Expand |z₁+z₂|² = (z₁+z₂)(z̄₁+z̄₂) and use z₁z̄₁ = |z₁|². The cross term 2Re(z₁z̄₂) is the complex analogue of the 2ab term in (a+b)². This identity is essential for proving the triangle inequalities.

The modulus of a sum never exceeds the sum of moduli. Geometrically: one side of a triangle is at most the sum of the other two. Equality holds iff z₁ and z₂ have the same argument — i.e., they point in the same direction from the origin.

The difference of moduli is at most the modulus of the difference. Gives a lower bound on |z₁−z₂|. Equality holds iff z₁ and z₂ are on the same ray from the origin (same argument). Combines with cn16 to bound |z₁+z₂| from below and above.

The argument is the angle θ that z makes with the positive real axis, measured anticlockwise. It is not uniquely defined — adding any multiple of 2π gives another valid argument. The set of all arguments is {θ + 2nπ : n ∈ ℤ}.

The principal value of the argument, denoted Arg(z) with a capital A, is the unique argument lying in (−π, π]. Not defined for z = 0. The interval is open at −π and closed at π — so exactly π is valid, but −π is not.

arctan(b/a) gives the correct angle only in Q1 and Q4. For Q2 and Q3 (a < 0), a correction of ±π is needed to land in (−π, π]. This is the practical formula for computing Arg(z) from coordinates.

Arguments add under multiplication and subtract under division — modulo 2π. Also: arg(z̄) = −arg(z), and arg(zⁿ) = n·arg(z). These hold as general arguments; for principal arguments, add 2kπ corrections as needed to stay in (−π, π].

Every complex number can be written in polar form using its modulus r and argument θ. This form makes multiplication and division geometric: multiply moduli and add arguments. The expression cos θ + i sin θ is often abbreviated as cis θ.

The most important formula connecting exponential and trigonometric functions. Derived by substituting iθ into the Taylor series of eˣ and separating real and imaginary parts. The special case θ = π gives Euler's identity: e^(iπ) + 1 = 0.

Compact form combining modulus and argument. Makes exponentiation and root extraction clean. Multiplication: r₁e^(iθ₁) · r₂e^(iθ₂) = r₁r₂ · e^(i(θ₁+θ₂)). This is the form used in De Moivre's theorem and all of M3.

Conjugation negates the argument; inversion negates the argument and reciprocates the modulus. Geometrically: conjugate reflects across the real axis; inverse reflects across the real axis and scales by 1/r².

Follows from z = re^(iθ) by taking ln of both sides: ln z = ln r + iθ. Since arg(z) is multi-valued, ln z is also multi-valued — the general value is ln|z| + i(Arg(z) + 2nπ). The principal value uses Arg(z) ∈ (−π, π].