The modulus of the product of two complex numbers z1 and z2 equals which of the following?

The modulus of the product of two complex numbers z1 and z2 equals which of the following?

• A. |z1 z2| = |z1| |z2|
• B. |z1 z2| = |z1| + |z2|
• C. |z1 z2| = |z1| - |z2|
• D. |z1 z2| = |z1| |z2|

Answer: (D)

When you multiply two complex numbers, their magnitudes multiply. Think of each complex number in polar form: z1 = r1 e^{iθ1} and z2 = r2 e^{iθ2}. Their product is z1 z2 = r1 r2 e^{i(θ1+θ2)}, so the modulus is r1 r2. Since r1 = |z1| and r2 = |z2|, you get |z1 z2| = |z1| |z2|. Geometrically, multiplication scales by the second magnitude and rotates, so the distance from the origin multiplies. For a quick check, if z1 has modulus 5 and z2 has modulus √2, the product has modulus 5√2, exactly the product of the moduli. The other options would imply adding or subtracting magnitudes, which isn’t how complex multiplication works.

When you multiply two complex numbers, their magnitudes multiply. Think of each complex number in polar form: z1 = r1 e^{iθ1} and z2 = r2 e^{iθ2}. Their product is z1 z2 = r1 r2 e^{i(θ1+θ2)}, so the modulus is r1 r2. Since r1 = |z1| and r2 = |z2|, you get |z1 z2| = |z1| |z2|. Geometrically, multiplication scales by the second magnitude and rotates, so the distance from the origin multiplies. For a quick check, if z1 has modulus 5 and z2 has modulus √2, the product has modulus 5√2, exactly the product of the moduli. The other options would imply adding or subtracting magnitudes, which isn’t how complex multiplication works.