1. The Complex Plane & the Argand Diagram
Historical Note: The Argand diagram is named after Jean-Robert Argand (1806), though it was actually first described by the Norwegian-Danish mathematician Caspar Wessel. It made complex numbers geometrically visible and helped "imaginary" numbers become accepted in mainstream mathematics.
1.1 Complex Numbers as Points
Every complex number $z = a + bi$ corresponds to a unique point $(a,\, b)$ in a two-dimensional coordinate plane called the complex plane (or Argand plane).
- The horizontal axis is the real axis (labelled $\text{Re}$). It carries all real numbers.
- The vertical axis is the imaginary axis (labelled $\text{Im}$). It carries all purely imaginary numbers.
- The real part $a$ gives the horizontal coordinate; the imaginary part $b$ gives the vertical coordinate.
Key Convention: When you plot $z = 3 + 4i$, you go to the point $(3,\, 4)$. The $i$ is not part of the coordinate — you use only the numerical coefficient $4$, not "$4i$".
1.2 The Position-Vector Reading
Each complex number $z = a + bi$ can equally be viewed as the position vector $\vec{OZ}$ from the origin $O = (0,0)$ to the point $Z = (a,b)$. This dual reading — point or vector — is exactly the same idea you met with position vectors in coordinate geometry, and we will use whichever reading is more convenient throughout this lesson.
| Complex Number $z$ | Point $(x,y)$ | Quadrant / Axis |
|---|---|---|
| $3 + 2i$ | $(3, 2)$ | Quadrant I |
| $-4 + i$ | $(-4, 1)$ | Quadrant II |
| $-2 - 3i$ | $(-2, -3)$ | Quadrant III |
| $5 - 2i$ | $(5, -2)$ | Quadrant IV |
| $4$ (i.e. $4 + 0i$) | $(4, 0)$ | Real axis |
| $-3i$ (i.e. $0 - 3i$) | $(0, -3)$ | Imaginary axis |
| $0$ | $(0, 0)$ | Origin |
Interactive Argand Diagram
Interactive visualization — available in the full lesson
Each complex number is shown as a point and as a position vector from the origin.