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Complex numbers
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A Geometry & Algebra set, untimed
Named traps
7 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Modulus without square rootAdds real and imaginary parts (or skips the square root) instead of √a2+b2.→
- Polar power modulus not raisedRaises only the angle, forgetting the modulus is also raised to the power (De Moivre).→
- Radius not subtracted in min distanceComputes distance to the center but forgets to subtract the circle radius.→
- Real parameter treated complexTreats a real parameter λ as complex, giving the wrong locus/line.→
- Reflection negates wrong partNegates the real part instead of the imaginary part when reflecting across the real axis.→
- Rotation factor i vs -iUses the wrong rotation factor (i vs -i) for clockwise vs counterclockwise rotation.→
- |a+b| split as |a|+|b|Treats the modulus of a sum as the sum of moduli.→
Key formulas
6- z = a + bi, i2 = −1, a,b ∈ RRe(z)=a, Im(z)=b
- Modulus: |z| = √a2 + b2Conjugate: z̄ = a − bi
- Multiplication(a1+b1 i)(a2+b2 i) = (a1a2 − b1b2) + (a1b2 + a2b1)i
- Divisionmultiply numerator and denominator by the conjugate of the denominator
- Polar form: z = r(cos θ + i sin θ)De Moivre: zn = rn(cos nθ + i sin nθ)
- Rotation by ±π/2 about origin corresponds to multiplication by ±i
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations