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

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

• 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.

• 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.

• 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.

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

• 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.

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

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

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

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

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

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

• 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.

• Wave mechanicsRelated: Quantum wavefunctions generally take complex values.

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

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

• 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.

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