If z is any complex number satisfying |z-1|=1, then which of...

If $z$ is any complex number satisfying $|z-1|=1$, then which of the following is correct?
(a) $\arg (z-1)=2 \arg (z)$
(b) $2 \arg (z)=\frac{2}{3} \arg \left(z^{3}-4\right)$
(c) $\arg (z+1)=\arg (z-1)$
(d) $2 \arg (z+1)=\arg (z-1)$

Key Concepts

Complex Numbers

Complex numbers, expressed as z = x + yi, form the foundation of complex analysis and algebra. They allow the representation of numbers that include both real and imaginary parts, bridging algebraic operations with geometric interpretations in the complex plane.

Polar Representation and Argument

Every non-zero complex number can be expressed in polar form as z = r e^(i?), where r is the modulus (or absolute value) and ? is the argument (or angle). This representation enables a clear geometric interpretation, particularly in understanding rotations and dilations, and is fundamental when manipulating arguments algebraically.

Modulus and Circle Equations in the Complex Plane

The modulus of a complex number provides a measure of distance from the origin or another point in the complex plane. Equations of the form |z - a| = r define circles centered at the point a with radius r, linking algebraic expressions directly to geometric figures.

Properties of the Argument Function

The argument of a complex number expresses the angle between the positive real axis and the line representing the complex number in the plane. Key properties include the behavior of arguments under multiplication and division, as well as the effects of translation. Understanding these properties is crucial for analyzing relationships between the arguments of different complex expressions.

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