Complex numbers
54 problems · Complex plane, De Moivre's formula, roots of unity and complex functions.
Ex.190A power of 1 + i
Complex numbers
L1
Ex.220Basic operations on complex numbers
Complex numbers
High school
Ex.221The equation $z=\bar z$
Complex numbers
High school
Ex.222Modulus via the conjugate
Complex numbers
High school
Ex.421Midpoint and centroid in the complex plane
Complex numbers
High school
Ex.422Argument of a quotient
Complex numbers
L1
Ex.423Properties of the conjugate
Complex numbers
High school
Ex.424The cube roots of unity
Complex numbers
High school
Ex.425De Moivre's formula
Complex numbers
High school
Ex.426Exponential form
Complex numbers
High school
Ex.427Geometric interpretation of operations
Complex numbers
High school
Ex.428Rotation via multiplication
Complex numbers
High school
Ex.429Modulus and argument
Complex numbers
High school
Ex.430Modulus of a product
Complex numbers
L1
Ex.431A quadratic with complex roots
Complex numbers
High school
Ex.432Product of complex numbers in trigonometric form
Complex numbers
High school
Ex.433Solving $z^3=1$
Complex numbers
High school
Ex.825Argument of a product
Complex numbers
L1
Ex.826Conjugate roots of real polynomials
Complex numbers
High school
Ex.827Cube roots of a complex number
Complex numbers
L1
Ex.828Linearizing powers of cosine
Complex numbers
High school
Ex.829De Moivre's formula by induction
Complex numbers
L1
Ex.830A circle given by a modulus equation
Complex numbers
L1
Ex.831The modulus equation $|z-a|=|z-b|$
Complex numbers
L1
Ex.832Triangle inequality for complex numbers
Complex numbers
High school
Ex.833A locus defined by a modulus equation
Complex numbers
L1
Ex.834The $n$-th roots of unity
Complex numbers
L1
Ex.835The perpendicular bisector
Complex numbers
L1
Ex.836Square roots of a complex number
Complex numbers
L1
Ex.837Computing roots with De Moivre
Complex numbers
L1
Ex.838Sum of the $n$-th roots of unity
Complex numbers
L1
Ex.839Linearizing $\sin^4\theta$
Complex numbers
L1
Ex.1227The sum of 1/(1 − ω^k)
Complex numbers
Olympiad
Ex.1229Three unit complex numbers with zero sum
Complex numbers
Olympiad
Ex.1294Three complex numbers of modulus one
Complex numbers
Olympiad
Ex.1305Sums over roots of unity from a logarithmic derivative
Complex numbers
Olympiad
Ex.1567An inequality involving arguments
Complex numbers
Olympiad
Ex.1568An ellipse via a modulus equation
Complex numbers
Olympiad
Ex.1569Equilateral triangle condition
Complex numbers
Olympiad
Ex.1570A locus involving an angle
Complex numbers
Olympiad
Ex.1571A region in the complex plane
Complex numbers
Olympiad
Ex.1572The modulus of a sum
Complex numbers
Olympiad
Ex.1573Product formula for cosines
Complex numbers
Olympiad
Ex.1574Sum of a trigonometric series
Complex numbers
Olympiad
Ex.1575Square roots of complex numbers
Complex numbers
L1
Ex.1576Sum of cosines
Complex numbers
Olympiad
Ex.1577Trigonometric identities via complex numbers
Complex numbers
Olympiad
Ex.1932Binomial coefficients in steps of three
Complex numbers
Olympiad
Ex.1937The product of sin(kπ/n)
Complex numbers
Olympiad
Ex.2103Solving a trigonometric equation via cubics
Complex numbers
Olympiad
Ex.2104The Julia set of $z\mapsto z^2$
Complex numbers
Olympiad
Ex.2105The Möbius transformation
Complex numbers
M1
Ex.2106A product formula for sines
Complex numbers
Olympiad
Ex.2107Integrating over the unit circle
Complex numbers
Olympiad
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