Complex Numbers

Definition and Standard Form

Complex numbers extend the real number system by introducing the imaginary unit. They are fundamental in precalculus and higher mathematics.

• Imaginary Unit: The imaginary unit, denoted as i, is defined by the property .

• Complex Number: Any number of the form , where and are real numbers and is the imaginary unit, is called a complex number.

• Standard Form: The standard form of a complex number is .

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using rules similar to those for binomials, with special attention to the property .

• Addition/Subtraction: Combine like terms: .

• Multiplication: Use distributive property and :

Complex Conjugate

The complex conjugate of a complex number is . The conjugate is useful for simplifying expressions and dividing complex numbers.

• Definition: For , the conjugate is .

• Product with Conjugate: The product of a complex number and its conjugate is always a real number:

• Example: The conjugate of is . Their product is .

Division of Complex Numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to obtain a real number in the denominator.

• Process:

  1. Given , multiply numerator and denominator by .

  2. Simplify numerator and denominator using .

  3. Express the result in standard form .

• Formula:

• Example:

  • Multiply numerator and denominator by :

  • Numerator:

  • Denominator:

  • Result:

Principal Square Root of Negative Numbers

For any positive real number , the principal square root of the negative number is defined using the imaginary unit:

• Example:

Summary Table: Properties of Complex Numbers