Complex Numbers

Introduction to Complex Numbers

Complex numbers extend the real number system by including imaginary numbers. A complex number is written in the standard form:

• Standard Form: , where a is the real part and b is the imaginary part.

• Imaginary Unit: is defined such that .

Example: has real part 3 and imaginary part 2.

Identifying Real and Imaginary Parts

Each complex number can be separated into its real and imaginary components.

• Real Part: The coefficient of 1 ( in ).

• Imaginary Part: The coefficient of ( in ).

Example: For , real part is 4, imaginary part is -2.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary).

• Addition:

• Subtraction:

Example:

Multiplying Complex Numbers

Multiplication uses the distributive property (FOIL method), remembering that .

• Formula:

• Simplify using .

Example:

Complex Conjugates

The conjugate of a complex number is . Conjugates are useful for rationalizing denominators and always yield a real number when multiplied together.

• Conjugate: If , then .

• Product:

Example: The conjugate of is .

Dividing Complex Numbers

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator to eliminate from the denominator.

• Formula:

• Simplify denominator to .

Example: : Multiply numerator and denominator by to get

Summary Table: Complex Number Operations

Additional info: These notes cover the foundational operations with complex numbers, which are essential for Precalculus topics such as polynomial equations, rational functions, and further algebraic manipulations.