Complex Numbers and the Complex Plane

Complex Numbers and the Complex Plane

Plotting Complex Numbers

Complex numbers can be represented as points in the complex plane, where the horizontal axis (real axis) represents the real part and the vertical axis (imaginary axis) represents the imaginary part of the number.

• Complex Number: A number of the form $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).

• Real Part: The value $a$ in $z = a + bi$.

• Imaginary Part: The value $b$ in $z = a + bi$.

• Complex Plane: A coordinate system where the x-axis is the real axis and the y-axis is the imaginary axis.

Example: Plotting $3 + 4i$

• Locate $3$ on the real axis (horizontal).

• Locate $4$ on the imaginary axis (vertical).

• The point $(3, 4)$ represents the complex number $3 + 4i$.

Formula:

Any complex number $z = a + bi$ is plotted at the point $(a, b)$ in the complex plane.

Application: Plotting complex numbers is foundational for understanding their geometric interpretation, which is essential for operations such as addition, subtraction, and finding the modulus or argument of a complex number.