This HL mini-lesson covers Complex Numbers: Cartesian arithmetic and the conjugate, modulus & argument on the Argand diagram, polar and Euler form, de Moivre's theorem, and roots of complex numbers and polynomials.
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With i² = −1, a complex number is z = a + bi (a = Re z, b = Im z). Real numbers are the special case b = 0.
Add/subtract componentwise; multiply with i² = −1. The conjugate of a + bi is a − bi.
Division: multiply top and bottom by the conjugate of the denominator to make it real.
Plot z = a + bi as the point (a, b). Its distance from the origin is the modulus, and the angle from the positive real axis is the argument.
Any complex number can be written using its modulus r and argument θ:
Tap an expression, then what kind of number it is.
Euler's relation links exponentials and trigonometry:
Raising a polar number to a power multiplies the modulus and the argument:
Tap a statement on the left, then its matching partner on the right.
The n nth roots of a complex number are equally spaced on a circle of radius |z|^{1/n}, separated by 2π/n.
Conjugate root theorem: a polynomial with real coefficients has non-real roots in conjugate pairs (if a + bi is a root, so is a − bi).
Use Cartesian form for adding, polar/Euler form for multiplying, dividing and taking powers or roots (via de Moivre). Convert between forms with |z| and arg z.
Exam habit: give arguments in the correct range and remember the modulus of a product is the product of the moduli.
Cartesian: z = a + bi, i² = −1; conjugate a − bi
Modulus/argument: |z| = √(a²+b²), arg z = arctan(b/a)
Polar/Euler: z = r cis θ = r e^{iθ}
de Moivre: (r cis θ)ⁿ = rⁿ cis(nθ)
Roots: n nth-roots on a circle; real polys have conjugate-pair roots
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