Introduction to Complex Numbers

Complex numbers are numbers of the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

Visualizing Complex Numbers

Every complex number can be represented as a point on the complex plane (also called the Argand diagram). The horizontal axis represents the real part \( a \), and the vertical axis represents the imaginary part \( b \). This allows us to use geometry to understand operations on complex numbers.

Polar Form

A complex number can also be written in polar form: \[ z = r(\cos \theta + i\sin \theta) \] where \( r \) is the modulus (distance from the origin) and \( \theta \) is the argument (angle with the real axis).

Why Geometry Matters

Geometric interpretations make it easier to understand operations like addition, multiplication, and complex conjugation. For instance, multiplying two complex numbers rotates and scales them in the plane.

Key Points

• Complex numbers extend the real numbers and provide solutions to equations like \( x^2 + 1 = 0 \).
• The geometric view helps in visualizing and understanding their behavior.