Complex Plane Operations

Just like real numbers can be placed on a 1D number line, complex numbers can be plotted on a 2D coordinate system known as the complex plane (or Argand diagram). This visual representation makes it much easier to understand the magnitude and behavior of complex numbers.

The Complex Plane

A complex number is written in the standard form $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part. In the complex plane:

• The horizontal axis is the Real Axis (representing $a$).
• The vertical axis is the Imaginary Axis (representing $b$).

To plot a complex number $z = a + bi$, you simply plot the coordinate point $(a, b)$. For example, to plot $3 - 2i$, you move $3$ units to the right along the real axis and $2$ units down along the imaginary axis.

The Modulus (Distance from the Origin)

The modulus (or absolute value) of a complex number is its straight-line distance from the origin $(0,0)$. It is denoted by $|z|$.

Because the real and imaginary components form a right triangle with the origin, we can use the Pythagorean theorem to find this distance. The modulus of $z = a + bi$ is calculated as: $|z| = \sqrt{a^2 + b^2}$

Note: Do not include the $i$ in this formula. You are squaring the real number $b$, not $bi$.

Complex Conjugates

The complex conjugate of $a + bi$ is $a - bi$. Geometrically, finding the conjugate means reflecting the point across the horizontal real axis. Because the triangle's dimensions remain the same, a complex number and its conjugate always have the exact same modulus.

Geometric Addition (Parallelogram Rule)

When you add two complex numbers algebraically, you combine their real parts and their imaginary parts separately.

Geometrically, this works exactly like vector addition. If you plot two complex numbers as arrows starting from the origin, you can find their sum by drawing a parallelogram. The diagonal of the parallelogram starting from the origin represents the sum of the two complex numbers.

Example Problems

Example 1: Plot $3 - 2i$ on the complex plane and find its distance from the origin.

• Plotting: Start at the origin, move $3$ units to the right on the real axis, and $2$ units down on the imaginary axis to point $(3, -2)$.
• Distance (Modulus): $|3 - 2i| = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$

The distance from the origin is $\sqrt{13}$.

Example 2: Find $|4 + 3i|$ and graph $4 + 3i$ and its conjugate.

• Modulus: $|4 + 3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
• Graphing: The number $4 + 3i$ is plotted at $(4, 3)$. Its conjugate is $4 - 3i$, which is plotted at $(4, -3)$. If you look at the complex plane, $(4, -3)$ is simply a mirror reflection of $(4, 3)$ across the horizontal real axis!