In polar form, the product and quotient of two complex numbers are obtained geometrically rather than algebraically. The magnitude of the product equals the product of the input magnitudes, and its argument equals the sum of the input arguments; division reverses the latter operation. The constructions that follow develop these rules on the Argand plane and provide the geometric foundation for evaluating a transfer function $H(s)$ from its pole and zero locations.
A complex number expressed in polar form is written $z = r\,e^{i\theta}$, where $r = |z|$ denotes its magnitude and $\theta = \arg z$ denotes its argument, measured anticlockwise from the positive real axis. Application of the angle-addition identity to the product of two such numbers yields:
The product has magnitude $r_1 r_2$ and argument $\theta_1 + \theta_2$. Multiplication by $z_2$ therefore acts on $z_1$ as the composition of two geometric operations on the plane: a rotation through the angle $\theta_2$ and a scaling by the factor $r_2$. These operations commute, so the order in which they are applied does not affect the result.
The arc from the positive real axis to $z_1$ subtends the angle $\theta_1$, and the arc to $z_2$ subtends $\theta_2$. The arc to the product subtends $\theta_1 + \theta_2$, equal to the two preceding arcs placed end to end. The length of the product vector equals $r_1 r_2$. When both inputs lie on the unit circle, the product also lies on the unit circle, and multiplication reduces to a pure rotation.
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Division of complex numbers in polar form follows from the same identity applied to the reciprocal:
The quotient has magnitude $r_1 / r_2$ and argument $\theta_1 - \theta_2$. Division by $z_2$ is the inverse of multiplication by $z_2$, comprising a rotation through $-\theta_2$ and a scaling by $1/r_2$. Composition of the two operations returns $z_1$ to itself, since $z_1 \cdot z_2 \cdot z_2^{-1} = z_1$.
The arc to the quotient subtends $\theta_1 - \theta_2$. When $\theta_2 > \theta_1$, this difference is negative and the quotient lies below the positive real axis even when both inputs lie above it. The length of the quotient vector equals $r_1$ scaled by the factor $1/r_2$. A divisor of magnitude greater than one therefore contracts the result, while a divisor of magnitude less than one expands it. A divisor that lies on the unit circle preserves the magnitude and contributes only a rotation.
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Multiplication by a fixed complex number $a$ defines a linear transformation of the plane that maps the unit vector at $1$ to $a$. Every point $w$ is sent to $aw$, so the rotation and scaling applied to the unit vector are applied uniformly to the entire Cartesian grid.
Setting $z_1$ and $z_2$ and dragging the transform handle away from $1$ updates the overlay. When the handle is placed on $z_1$, the overlay displays the grid transformation associated with multiplication by $z_1$, and the point $z_2$ is mapped to the corresponding product. Selecting division displays the inverse transformation, in which each point is rotated through $-\arg a$ and scaled by $1/|a|$.