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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.

• Real numberCompared with: Complex numbers extend the reals with values that do not lie on the real line.

• Carl Friedrich GaussRelated: Gauss helped establish complex numbers as a coherent mathematical system.

• IntegerCompared with: Complex numbers extend beyond the real line that contains the integers.

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

• Finite fieldCompared with: The complex numbers are algebraically closed and infinite, unlike finite fields.

• 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.

• Quantum computingRelated: Quantum amplitudes are complex numbers whose magnitudes and phases shape outcomes.

• Born ruleRelated: Quantum amplitudes can be complex, though their squared magnitudes are real.

• ZeroRelated: Zero also exists in the complex numbers, where it is the additive identity.

• Cross productCompared with: In two dimensions, complex multiplication can encode rotation and scaling without a vector cross product.

• QuaternionRelated: Quaternions extend complex numbers with two additional imaginary units.

• Ordered pairRelated: Its real and imaginary parts can be represented as an ordered pair of real numbers.

• VectorCompared with: Complex numbers can be drawn as plane arrows, but their algebraic role differs from general vectors.

• Quadratic equationRelated: Some quadratic equations have no real roots but do have complex solutions.

• SineRelated: Complex arguments extend sine beyond real angles.

• Square rootRelated: Negative real numbers have square roots only after the number system is extended to complex numbers.

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

• Trigonometric functionsRelated: Complex inputs extend trigonometric functions beyond real angles and introduce hyperbolic behavior.

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

• Absolute convergenceRelated: For complex series, absolute values measure magnitudes and still guarantee convergence.

• Normed vector spaceRelated: Complex scalars require absolute-value homogeneity in the norm axioms.

• 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.

• Cyclotomic polynomialRelated: The roots defining cyclotomic polynomials lie on the complex unit circle.

• 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.

• p-adic numberCompared with: The complex field extends the real completion, while p-adic fields arise from prime-based absolute values.

• 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.

• Algebraic closureCompared with: The complex field is algebraically closed, making it a familiar example of an algebraic closure of the reals.

• 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.

• FieldBroader topic: The complex numbers form an algebraically closed field.

• Polynomial equationRelated: Allowing complex values supplies roots that may not exist among the reals.

• Negative numberCompared with: Negative real values lie on the real axis, unlike non-real complex values.

• 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.

• Quadratic formulaRelated: When the discriminant is negative, the square root in the formula is complex.