Argand Diagram
The Argand diagram (also called the complex plane) is a geometric representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Definition
A complex number $z = a + ib$ is plotted as the point $(a, b)$ on the Argand diagram.
Components
- Horizontal axis: Real axis
- Vertical axis: Imaginary axis
- Modulus ($r$): The distance from the origin to the point $(a, b)$, given by $r = |z| = \sqrt{a^2 + b^2}$
- Argument ($\theta$): The angle measured from the positive real axis to the line segment joining the origin to the point $(a, b)$, given by $\theta = \tan^{-1}\left(\frac{b}{a}\right)$
- Principal argument: $-\pi < \theta \leq \pi$
Complex Conjugate on the Argand Diagram
The complex conjugate $\overline{z} = a - ib$ is the reflection of $z = a + ib$ across the real axis.