Trap 1: sqrt(-a) * sqrt(-b) = sqrt(ab) WRONG. sqrt(-a) * sqrt(-b) = isqrt(a) * isqrt(b) = -sqrt(ab). The product rule for square roots fails for negative numbers.
Trap 2: arg(z1+z2) = arg(z1) + arg(z2) WRONG. Argument adds for multiplication, NOT addition.
Trap 3: Conjugate root theorem for complex coefficients Only works for REAL coefficients. Complex-coefficient polynomials don't have conjugate pairs.
Trap 4: Forgetting quadrant in argument arctan(1/sqrt(3)) = pi/6 for ANY quadrant, but the principal argument differs: pi/6 (Q1), 5pi/6 (Q2), -5pi/6 (Q3), -pi/6 (Q4).
Trap 5: |z1+z2| = |z1|+|z2| always Only when they point in the same direction. Otherwise strict inequality.
Trap 6: |z-z0| < r means circle No, < (strict) means the open disk (interior). = means the circle itself.
Trap 7: arg(-1) = -pi Wrong. arg(-1) = pi. The principal value convention includes pi but excludes -pi.
Trap 8: Complex numbers can be ordered False. There is no total ordering on C compatible with arithmetic. We can only compare moduli.
Exam checklist: Before finalizing: (1) Did I check the quadrant? (2) Is my argument in (-pi, pi]? (3) Did I use z*z-bar = |z|^2 correctly? (4) For cube roots, did I use 1+w+=0?