The modulus of the quotient z1/z2 equals which expression?

The modulus of the quotient z1/z2 equals which expression?

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

Answer: (B)

The size (modulus) of a complex number behaves nicely under division: the modulus of a quotient equals the quotient of the moduli. In symbols, for z1 and z2 with z2 ≠ 0, |z1/z2| = |z1|/|z2|. This follows from writing the numbers in polar form z1 = |z1| e^{iθ1} and z2 = |z2| e^{iθ2}, so z1/z2 = (|z1|/|z2|) e^{i(θ1−θ2)}, whose magnitude is |z1|/|z2|. The other expressions would describe the modulus of a product or simple sums/differences of moduli, which do not generally match the modulus of the quotient.

The size (modulus) of a complex number behaves nicely under division: the modulus of a quotient equals the quotient of the moduli. In symbols, for z1 and z2 with z2 ≠ 0, |z1/z2| = |z1|/|z2|. This follows from writing the numbers in polar form z1 = |z1| e^{iθ1} and z2 = |z2| e^{iθ2}, so z1/z2 = (|z1|/|z2|) e^{i(θ1−θ2)}, whose magnitude is |z1|/|z2|. The other expressions would describe the modulus of a product or simple sums/differences of moduli, which do not generally match the modulus of the quotient.