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

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

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

Answer: (D)

When a complex number is written in polar form z = r e^{iθ}, dividing two numbers multiplies their moduli and subtracts their angles: z1/z2 = (r1/r2) e^{i(θ1 − θ2)}. So the angle or argument of the quotient is the difference of the angles: arg(z1/z2) = θ1 − θ2 = arg(z1) − arg(z2), up to adding multiples of 2π (since angles are defined modulo 2π). This is why the correct expression is arg z1 minus arg z2. The other possibilities would correspond to different operations: the angle of a product is the sum of the angles, not the quotient; and taking the ratio of the arguments isn’t a standard way to combine angles. Also remember z2 must be nonzero for the quotient to be defined.

When a complex number is written in polar form z = r e^{iθ}, dividing two numbers multiplies their moduli and subtracts their angles: z1/z2 = (r1/r2) e^{i(θ1 − θ2)}. So the angle or argument of the quotient is the difference of the angles: arg(z1/z2) = θ1 − θ2 = arg(z1) − arg(z2), up to adding multiples of 2π (since angles are defined modulo 2π). This is why the correct expression is arg z1 minus arg z2.

The other possibilities would correspond to different operations: the angle of a product is the sum of the angles, not the quotient; and taking the ratio of the arguments isn’t a standard way to combine angles. Also remember z2 must be nonzero for the quotient to be defined.