Equations and polynomials: complex roots, multiplicities and factorization
From quadratic equations to complex factors: repeated roots, conjugates, the fundamental theorem of algebra and Horner division. Examples, exercises and a lab.
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Clear, organized learning paths that build strong foundations in mathematics and computer science.
Complex numbers
Complex roots: all solutions of zⁿ = w
All roots, regular polygons and the principal root: derived formulas, worked examples, exercises and an interactive lab.
ReadComplex numbers, stage 5: Euler, exponential form and De Moivre
From e^(iθ)=cosθ+i sinθ to De Moivre: why it works, positive and negative integer powers and what approximations show. Examples, twelve exercises and an interactive lab.
ReadComplex numbers, stage 4: trigonometric form, rotations and scaling
Why does multiplying a complex number rotate and scale it? Coordinates, sine and cosine, proof of the product rule, division, centres beyond the origin, examples and an animated lab.
ReadComplex numbers, stage 3: modulus, distance and argument
Read a+bi as a point: Pythagoras, distance between complex numbers, degrees and radians, principal argument and loci. Examples, exercises and a draggable-point lab.
ReadComplex numbers, stage 2: calculating with a+bi
Sum, difference, product, conjugate, division and powers of i: derived formulas, worked examples, exercises with hidden solutions and a lab using exact fractions.
ReadComplex numbers, stage 1: why they arise and how to represent them
From x²+1=0 to the imaginary unit i, the form a+bi and the complex plane: guided examples, twelve exercises and an interactive lab.
ReadComplex numbers: a path from first steps to applications
An eight-stage path to understand complex numbers, calculate and interpret them in the plane. Stages 1, 2, 3, 4, 5, 6 and 7, with their labs, are available.
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