The Geometric Interpretation of Complex Numbers
Every complex number $z = x + yi$ can be represented as a point in the complex plane, with the real part $x$ on the horizontal axis and the imaginary part $y$ on the vertical axis. This geometric picture makes the algebra of complex numbers visible and intuitive.
Quick Reference
| Concept | Description |
|---|---|
| Complex plane (Argand plane) | The $xy$-plane used to represent complex numbers |
| Real axis | The $x$-axis; represents the real part $\operatorname{Re}(z)$ |
| Imaginary axis | The $y$-axis; represents the imaginary part $\operatorname{Im}(z)$ |
| Point $z = x + yi$ | Plotted at coordinates $(x, y)$ |
| $z_1 + z_2$ | Fourth vertex of the parallelogram with vertices $O$, $z_1$, $z_2$ |
| $z_1 - z_2$ | Arrow from $z_2$ to $z_1$; the other diagonal of the parallelogram |
History and Motivation
In the 16th century, mathematicians began using square roots of negative numbers to solve quadratic and cubic equations. However, they struggled to find a clear explanation for these expressions, which were often met with skepticism. The term "imaginary" arose from the belief that these numbers were abstract or unreal.
As time went on and mathematicians discovered how powerful these numbers are, a simple geometric interpretation emerged in the 19th century. This visual approach made complex numbers far easier to understand and removed lingering doubts about their legitimacy.
In modern mathematics, the geometric interpretation is not logically necessary, since the formal definitions of addition and multiplication are sufficient. But the visual representation remains extremely useful for building intuition.
The Complex Plane
We represent a complex number $z = x + yi$ as a point in the plane, where:
- The horizontal axis (the real axis) represents the real part $\operatorname{Re}(z) = x$.
- The vertical axis (the imaginary axis) represents the imaginary part $\operatorname{Im}(z) = y$.
This plane is called the complex plane, or the Argand plane (after Jean-Robert Argand), or the $z$-plane.
Special positions in the complex plane:
- Real numbers ($b = 0$): $z = x + 0i$ lies on the real axis.
- Purely imaginary numbers ($a = 0$): $z = 0 + yi$ lies on the imaginary axis.
It is common practice to use the terms "complex number" and "point" interchangeably. We often say "the point $z
quot; as shorthand for "the point corresponding to the complex number $z$."
Geometric Addition: The Parallelogram Law
Let $z_1 = a + bi$ and $z_2 = c + di$ be two complex numbers plotted in the complex plane. Their sum is:
$ z_1 + z_2 = (a + c) + (b + d)i $Parallelogram Law. When $z_1$ and $z_2$ are plotted as points (or arrows from the origin), their sum $z_1 + z_2$ is the fourth vertex of the parallelogram whose other three vertices are the origin $O$, $z_1$, and $z_2$.
Geometric Subtraction
Since $z_1 - z_2 = z_1 + (-z_2)$, we can think of subtraction as adding $z_1$ to the opposite of $z_2$ (same length, opposite direction).
Using the parallelogram law with $z_1$ and $-z_2$, the result $z_1 - z_2$ is the arrow from $z_2$ to $z_1$. This arrow is parallel to, and the same length as, the other diagonal of the parallelogram formed by $O$, $z_1$, and $z_2$.
Example. Let $z_1 = 3 + i$ and $z_2 = 1 + 2i$. Plot both in the complex plane and find $z_1 + z_2$ and $z_1 - z_2$ geometrically.
Solution.
- $z_1 + z_2 = (3+1) + (1+2)i = 4 + 3i$. This is the fourth vertex of the parallelogram with vertices $O = (0,0)$, $z_1 = (3,1)$, $z_2 = (1,2)$.
- $z_1 - z_2 = (3-1) + (1-2)i = 2 - i$. This is the arrow from $z_2 = (1,2)$ to $z_1 = (3,1)$.
Frequently Asked Questions
What is the complex plane?
The complex plane is the ordinary $xy$-plane reinterpreted to represent complex numbers. The horizontal axis is the real axis (representing the real part of $z$), and the vertical axis is the imaginary axis (representing the imaginary part). Each complex number $z = x + yi$ corresponds to the point $(x, y)$. It is also called the Argand plane or the $z$-plane.
