| symbol | latex code | explanation | |
|---|---|---|---|
i | i | The imaginary unit, defined by i² = −1. It is the foundation of all imaginary numbers and complex arithmetic. | |
z | z | Standard variable representing a complex number, typically written as z = a + bi where a and b are real numbers. | |
a + bi | a + bi | Rectangular or algebraic form of a complex number, where a is the real part and b is the imaginary part. | |
Re(z) | \text{Re}(z) | The real part of z. For z = a + bi, Re(z) = a. Sometimes written as ℜ(z). | |
Im(z) | \text{Im}(z) | The imaginary part of z. For z = a + bi, Im(z) = b. Note that Im(z) itself is a real number. | |
ℂ | \mathbb{C} | The set of all complex numbers. Contains ℝ (real numbers) as a subset. | |
ℝ | \mathbb{R} | The set of all real numbers. In the complex plane, ℝ corresponds to the horizontal axis. | |
z̄ | \bar{z} | The complex conjugate of z. For z = a + bi, the conjugate is z̄ = a − bi. | |
z* | z^* | Alternative notation for the complex conjugate, commonly used in physics and engineering. | |
−z | -z | The additive inverse of z. For z = a + bi, we have −z = −a − bi. Satisfies z + (−z) = 0. | |
z⁻¹ | z^{-1} | The multiplicative inverse (reciprocal) of z. Satisfies z · z⁻¹ = 1. Equals z̄/|z|² for nonzero z. | |
1/z | \frac{1}{z} | Reciprocal of z, equivalent to z⁻¹. Computed by multiplying numerator and denominator by the conjugate. | |
z · z̄ = |z|² | z \cdot \bar{z} = |z|^2 | Fundamental identity: a complex number times its conjugate equals the square of its modulus. | |
|z| | |z| | The modulus (absolute value) of z, representing its distance from the origin in the complex plane. | |
|z| = √(a² + b²) | |z| = \sqrt{a^2 + b^2} | Formula for the modulus derived from the Pythagorean theorem. For z = 3 + 4i, |z| = 5. | |
arg(z) | \arg(z) | The argument of z — the angle from the positive real axis to z, measured counterclockwise. Multi-valued, differing by multiples of 2π. | |
Arg(z) | \text{Arg}(z) | The principal argument of z, restricted to (−π, π]. Gives a unique angle for each nonzero complex number. | |
θ = arctan(b/a) | \theta = \arctan\left(\frac{b}{a}\right) | Formula for the argument when a > 0. For other quadrants, adjustments of π are needed. | |
r(cos θ + i sin θ) | r(\cos\theta + i\sin\theta) | The trigonometric form of a complex number, where r = |z| and θ = arg(z). | |
r cis θ | r\,\text{cis}\,\theta | Shorthand for trigonometric form. "cis" stands for "cosine + i sine". | |
z = r∠θ | z = r\angle\theta | Polar notation commonly used in engineering, where r is the modulus and θ is the argument. | |
re^(iθ) | re^{i\theta} | The exponential form of a complex number, combining modulus r and argument θ. | |
e^(iθ) = cos θ + i sin θ | e^{i\theta} = \cos\theta + i\sin\theta | Euler's formula — the bridge between exponential and trigonometric forms. | |
e^(iπ) + 1 = 0 | e^{i\pi} + 1 = 0 | Euler's identity, connecting five fundamental constants: e, i, π, 1, and 0. | |
e^(a+bi) = eᵃ(cos b + i sin b) | e^{a+bi} = e^a(\cos b + i\sin b) | General formula for the complex exponential, separating the real exponential growth from the rotational component. | |
z₁ + z₂ | z_1 + z_2 | Addition of complex numbers: add real parts and imaginary parts separately. See operations. | |
z₁ − z₂ | z_1 - z_2 | Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i. | |
z₁ · z₂ | z_1 \cdot z_2 | Multiplication using FOIL and the fact that i² = −1. In polar form: multiply moduli, add arguments. | |
z₁/z₂ | \frac{z_1}{z_2} | Division: multiply numerator and denominator by the conjugate of the denominator. | |
|z₁z₂| = |z₁||z₂| | |z_1 z_2| = |z_1||z_2| | The modulus of a product equals the product of the moduli. | |
arg(z₁z₂) = arg(z₁) + arg(z₂) | \arg(z_1 z_2) = \arg(z_1) + \arg(z_2) | The argument of a product equals the sum of the arguments (modulo 2π). | |
zⁿ | z^n | The nth power of z. In exponential form: zⁿ = rⁿe^(inθ). | |
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) | (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) | De Moivre's theorem — essential for computing powers and roots of complex numbers. | |
zⁿ = rⁿe^(inθ) | z^n = r^n e^{in\theta} | Power formula in exponential form: raise the modulus to the nth power, multiply the argument by n. | |
ⁿ√z | \sqrt[n]{z} | The nth root of z. Has exactly n distinct values, evenly spaced around a circle in the complex plane. | |
ωₖ = e^(2πik/n) | \omega_k = e^{2\pi i k/n} | The nth roots of unity — solutions to zⁿ = 1. There are exactly n of them, for k = 0, 1, ..., n−1. | |
ω = e^(2πi/n) | \omega = e^{2\pi i/n} | The primitive nth root of unity. All other nth roots are powers of ω. | |
z = x + iy | z = x + iy | Complex number as a point (x, y) in the complex plane, also called the Argand diagram. | |
|z − z₀| = r | |z - z_0| = r | Equation of a circle with center z₀ and radius r in the complex plane. | |
|z − z₀| < r | |z - z_0| < r | Open disk: all points strictly inside the circle centered at z₀ with radius r. | |
|z − z₀| ≤ r | |z - z_0| \leq r | Closed disk: all points on or inside the circle centered at z₀ with radius r. | |
Im(z) > 0 | \text{Im}(z) > 0 | The upper half-plane — all complex numbers with positive imaginary part. | |
Re(z) > 0 | \text{Re}(z) > 0 | The right half-plane — all complex numbers with positive real part. | |
|z₁ + z₂| ≤ |z₁| + |z₂| | |z_1 + z_2| \leq |z_1| + |z_2| | Triangle inequality: the modulus of a sum is at most the sum of the moduli. See properties. | |
||z₁| − |z₂|| ≤ |z₁ − z₂| | ||z_1| - |z_2|| \leq |z_1 - z_2| | Reverse triangle inequality: useful for establishing lower bounds on moduli. | |
z + z̄ = 2Re(z) | z + \bar{z} = 2\text{Re}(z) | Sum of a complex number and its conjugate gives twice the real part. | |
z − z̄ = 2i Im(z) | z - \bar{z} = 2i\,\text{Im}(z) | Difference of a complex number and its conjugate gives twice the imaginary part times i. | |
|Re(z)| ≤ |z| | |\text{Re}(z)| \leq |z| | The absolute value of the real part never exceeds the modulus. | |
|Im(z)| ≤ |z| | |\text{Im}(z)| \leq |z| | The absolute value of the imaginary part never exceeds the modulus. | |
z² = w | z^2 = w | Quadratic equation in complex numbers — always has exactly two solutions (counting multiplicity). | |
zⁿ = w | z^n = w | The nth power equation has exactly n solutions in ℂ, evenly distributed around a circle. | |
az² + bz + c = 0 | az^2 + bz + c = 0 | Quadratic equation with complex coefficients. Solved using the quadratic formula with complex arithmetic. See equations and polynomials. | |
z = (−b ± √(b² − 4ac)) / 2a | z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} | Quadratic formula — works for all complex coefficients. The discriminant may be negative or complex. | |
Pₙ(z) = aₙzⁿ + ... + a₁z + a₀ | P_n(z) = a_n z^n + \cdots + a_1 z + a_0 | Polynomial of degree n with complex coefficients. By the Fundamental Theorem of Algebra, has exactly n roots in ℂ. | |
e^z | e^z | Complex exponential function. For z = a + bi: e^z = eᵃ(cos b + i sin b). | |
ln(z) | \ln(z) | Complex natural logarithm. Multi-valued: ln(z) = ln|z| + i(arg(z) + 2πk) for integer k. | |
Log(z) | \text{Log}(z) | Principal branch of the complex logarithm: Log(z) = ln|z| + i·Arg(z). | |
sin(z) | \sin(z) | Complex sine function, defined for all z ∈ ℂ. Unlike real sine, |sin(z)| can exceed 1. | |
cos(z) | \cos(z) | Complex cosine function. Related to exponentials: cos(z) = (e^(iz) + e^(−iz))/2. | |
sin(z) = (e^(iz) − e^(−iz))/2i | \sin(z) = \frac{e^{iz} - e^{-iz}}{2i} | Exponential definition of complex sine, derived from Euler's formula. | |
cos(z) = (e^(iz) + e^(−iz))/2 | \cos(z) = \frac{e^{iz} + e^{-iz}}{2} | Exponential definition of complex cosine, derived from Euler's formula. | |
sinh(z) | \sinh(z) | Hyperbolic sine: sinh(z) = (e^z − e^(−z))/2. Closely related to complex sine. | |
cosh(z) | \cosh(z) | Hyperbolic cosine: cosh(z) = (e^z + e^(−z))/2. Closely related to complex cosine. | |
sin(iz) = i sinh(z) | \sin(iz) = i\sinh(z) | Connection between trigonometric and hyperbolic functions through the imaginary unit. | |
cos(iz) = cosh(z) | \cos(iz) = \cosh(z) | Connection between complex cosine and hyperbolic cosine. |