Complex arithmetic, conjugates and modulus

Develop fluent complex arithmetic in standard form while keeping the meaning of each operation clear. This practice set covers real and imaginary parts, addition, subtraction, multiplication, conjugates, division and reciprocals. It uses the standard convention that the imaginary part is the real coefficient of the imaginary unit, rather than the full imaginary term.

Modulus connects the algebra to distance. You will calculate magnitudes, use the product of a number and its conjugate, find the distance between two complex points and test whether magnitudes can simply be added. Other questions solve a complex equation or determine a real parameter that makes a product real. Boundary cases include zero, purely real numbers and purely imaginary denominators; a reciprocal exists whenever the complex denominator is nonzero, even if one component vanishes.

Prerequisites are the power cycle of the imaginary unit, basic algebraic expansion, radicals and coordinate distance. Both single-select and multiselect questions progress from interpreting notation to multistep arithmetic and justification. Detailed explanations show how the real and imaginary components combine, how the defining square of the imaginary unit changes signs and how to check a quotient by multiplying back. Polar form and trigonometric methods are outside this set.