The Argand Diagram — Geometric Representation of Complex Numbers (9709 P3)

Core Concept

Setting Up the Argand Diagram

For a complex number $z = x + iy$:

• The real axis (horizontal) represents $\operatorname{Re}(z) = x$.
• The imaginary axis (vertical) represents $\operatorname{Im}(z) = y$.
• The point $P(x, y)$ represents $z$, and the position vector $\overrightarrow{OP}$ from the origin $O$ to $P$ also represents $z$.

The modulus $|z| = \sqrt{x^2 + y^2}$ is the distance from the origin to $P$. The argument $\arg(z)$ is the angle the vector $\overrightarrow{OP}$ makes with the positive real axis, measured anticlockwise, in the range $(-\pi, \pi]$.

Geometric Effects of Key Operations

Conjugation: The conjugate of $z = x + iy$ is $\bar{z} = x - iy$. Geometrically, this is a reflection in the real axis — the point moves from $(x, y)$ to $(x, -y)$.

Addition: To add $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$, components add: $z_1 + z_2 = (x_1+x_2) + i(y_1+y_2)$. Geometrically this is vector addition (the parallelogram or triangle rule): the vector representing $z_1 + z_2$ is the diagonal of the parallelogram formed by $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$.

Subtraction: $z_1 - z_2$ corresponds to adding $z_1$ and $-z_2$. The vector $\overrightarrow{P_2 P_1}$ (from $z_2$ to $z_1$) represents $z_1 - z_2$. This means $|z_1 - z_2|$ is the distance between the points representing $z_1$ and $z_2$ — a critical fact for loci questions.

Multiplication: Multiplying $z_1$ by $z_2$ gives a result with:

Geometrically, multiplying by $z_2$ scales the vector for $z_1$ by factor $|z_2|$ and rotates it anticlockwise by $\arg(z_2)$. A special case: multiplying by $i$ (which has modulus 1 and argument $\pi/2$) rotates any vector by $90°$ anticlockwise.

Division: Dividing $z_1$ by $z_2$ gives:

Geometrically this is a scaling by factor $\dfrac{1}{|z_2|}$ and a rotation clockwise by $\arg(z_2)$.