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

• 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.

• Helmholtz equationRelated: Complex-valued solutions compactly encode phase-shifted oscillations in wave problems.

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

• Software-defined radioRelated: Complex samples compactly encode the in-phase and quadrature components used in radio processing.

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

• Constructible numberRelated: Complex coordinates can also be constructed because the needed real and imaginary components are constructible.

• 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.

• Algebraically closed fieldBroader topic: The complex numbers are the standard example of an algebraically closed field.

• 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.

• ParenthesesRelated: Parentheses clarify signs and grouping when complex numbers appear in calculations.

• Partial fraction decompositionRelated: The chosen coefficient field affects which denominator factors count as irreducible.

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

• Wave mechanicsRelated: Quantum wavefunctions generally take complex values.

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

• Gerolamo CardanoRelated: Cubic solutions in Ars Magna exposed complex quantities even when the final answers were real.

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

• Magnitude (mathematics)Related: Its magnitude is the distance from the origin in the complex plane.

• Number (mathematics)Broader topic: Complex numbers extend the reals so equations such as x² = −1 have solutions.

• Cube rootRelated: Extending cube roots beyond real values introduces multiple complex roots.

• Euler's four-square identityRelated: The two-square norm of complex numbers provides a simpler model for multiplicative sums of squares.

• 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.

• Cubic functionRelated: A cubic can have nonreal roots, which occur as a conjugate pair.

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

• Sign (mathematics)Compared with: Complex numbers do not have the ordinary positive-or-negative ordering of real numbers.

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

• Gelfand–Mazur theoremBroader topic: The sole possible algebra is this field with its usual norm.

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

• Operator overloadingBroader topic: Arithmetic operators let complex numbers use familiar expressions such as addition and multiplication.