Multiplication in Polar Form

Complex Analysis · Axiom Academy

LESSON Multiplication in Polar Form Where complex multiplication becomes beautifully simple — multiply the lengths, add the angles. 1. The Rule: Multiply Moduli, Add Arguments Write each number by its length (modulus r ) and its direction (argument ). Then the product has length r_1 r_2 and direction — nothing else to track. Multiply the moduli, add the arguments 4 multiplications, then group and simplify the i^2 term. 1 multiplication and 1 addition. Start from the trigonometric form, multiply the two numbers out, and let the angle-addition identities for cosine and sine do the rest. Those identities are secretly the statement that arguments add. Apply angle-addition formulas: The cosine and sine angle-addition formulas are telling us that complex multiplication rotates and scales . Each factor contributes its own angle to the total rotation — so the arguments simply add. The rule isn't just bookkeeping — it describes a motion of the whole plane. Multiplying by does two things at once. Multiplying by a number of modulus r scales distances by r . If r > 1 points move outward; if r < 1 they move inward. Multiplying by a number of argument rotates everything counterclockwise by about the origin. So means: scale z_1 by r_2 , then rotate by . Order doesn't matter — the result is the same. Division reverses it: — divide the moduli, subtract the arguments.

This is the written version of the interactive lesson above. See the full Complex Analysis course.