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The 85 pages that link to Complex number, each with the reason it gives.

• Quantum mechanicsNarrower topic: Complex amplitudes allow the interference patterns central to quantum predictions.

• Schrödinger equationNarrower topic: Complex amplitudes let the equation encode phase as well as probability.

• Complex analysisNarrower topic: Complex-valued inputs and outputs are the basic objects of the subject.

• Complex planeNarrower topic: Each point is the geometric representation of one complex number.

• Natural numberNarrower topic: Complex numbers belong to a much broader number system than the natural numbers.

• Fundamental theorem of algebraNarrower topic: The theorem guarantees roots in this number system, even when real roots do not exist.

• Acoustic impedanceNarrower topic: Sinusoidal acoustic impedance is complex because pressure and flow can differ in phase.

• Euler's formulaNarrower topic: The identity equates two complex-number descriptions of the same value.

• WavefunctionNarrower topic: Wavefunctions can have complex values whose phases affect interference.

• Complex exponentialNarrower topic: Every input and output of the complex exponential is a complex number.

• Electrical impedanceNarrower topic: Complex numbers encode resistance and reactance together.

• PhasorNarrower topic: A phasor uses a complex number to store amplitude and phase together.

• Laplace transformNarrower topic: The transform variable is generally complex, and its real part controls exponential weighting.

• Root of unityNarrower topic: Complex numbers contain all roots of unity, including nonreal ones.

• Transcendental numberNarrower topic: Transcendence is defined for complex numbers as well as real numbers.

• Polynomial rootNarrower topic: Complex numbers provide a domain in which every nonconstant polynomial has roots.

• Gaussian integerNarrower topic: Gaussian integers restrict both real coordinates of complex numbers to integers.

• Möbius transformationNarrower topic: The input, output, and coefficients of a Möbius transformation are complex numbers.

• Wave functionNarrower topic: Wave-function amplitudes are generally complex, not merely real-valued.

• William Rowan HamiltonNarrower topic: Hamilton’s search for a three-dimensional analogue of complex numbers led to quaternions.

• Dirichlet seriesNarrower topic: The variable s ranges over complex numbers, making powers n⁻ˢ complex-valued.

• Fast Fourier transformNarrower topic: Fourier coefficients and phase factors are naturally expressed as complex values.

• Cauchy–Riemann equationsNarrower topic: The equations arise by expressing both input and output complex numbers in real coordinates.

• Complex conjugateNarrower topic: Conjugation acts on every complex number by preserving its real part and reversing its imaginary part.

• Complex multiplicationNarrower topic: Multiplication acts on these numbers and preserves their algebraic structure.

• Imaginary unitNarrower topic: The imaginary unit entered mathematics as the defining ingredient of a broader number system.

• Euler's identityNarrower topic: The exponent iπ makes the identity a statement about complex exponentiation.

• Euler's NumberNarrower topic: Extending exponentials to complex inputs reveals e's connection to rotation.

• Probability amplitudeNarrower topic: An amplitude's real and imaginary parts encode magnitude and phase.

• Complex modulusNarrower topic: The modulus assigns a nonnegative magnitude to every complex number.

• Cubic equationNarrower topic: Complex numbers provide a setting in which every cubic has three roots, counting multiplicity.

• Mandelbrot setNarrower topic: The parameter c and every iterated value belong to the complex plane.

• Scattering amplitudeNarrower topic: Amplitude phases affect interference even when probabilities use squared magnitudes.

• Abraham de MoivreNarrower topic: De Moivre’s formula connects powers of complex numbers to their trigonometric form.

• Beta functionNarrower topic: The beta function's arguments and values are commonly complex.

• Complex derivativeNarrower topic: The increment can approach zero through every direction in the complex plane.

• Polar formNarrower topic: Polar form is one way to represent this broader kind of number.

• Directed angleNarrower topic: Arguments of complex numbers encode oriented angles between rays from the origin.

• De Moivre's formulaNarrower topic: The formula applies to complex numbers expressed through their magnitude and angle.

• Nth rootNarrower topic: Complex numbers supply roots when a real number system has none and reveal multiple solutions.

• Complex conjugate root theoremNarrower topic: Nonreal roots live in the complex numbers, where conjugation creates the paired value.

• Imaginary numberNarrower topic: Imaginary numbers are precisely the complex numbers whose real component is zero.

• Gershgorin circle theoremNarrower topic: Disk centers and eigenvalues may both be complex, so the theorem's geometry lives in the complex plane.

• Casas-Alvero conjectureNarrower topic: The conjecture is posed over the complex numbers, where every nonconstant polynomial has a root.

• Marden's theoremNarrower topic: Cubic roots and derivative roots are complex numbers treated geometrically as points.