Divide Complex Numbers

Learning Outcomes

• Recall a technique: Rationalizing the denominator.

• Key Concept: Techniques effective in one mathematical situation can be effective in similar situations.

Dividing Complex Numbers

• Similar to how we rationalized denominators in fractions with radicals, we also employ a similar technique when dividing complex numbers.

• Complex Conjugate: An essential component in this method.

  • Definition: The complex conjugate of a complex number a + bi is a - bi.

• Objectives:

  • Identify and write the complex conjugate of a complex number.

  • Divide complex numbers.

  • Simplify powers of i.

Rationalizing the Denominator

• When rationalizing, we leverage the property that (√a)² = a.

• In complex division, we utilize the fact that i² = -1.

Complexity of Dividing Complex Numbers

• Division of complex numbers is inherently more complex than addition, subtraction, and multiplication because:

  • You cannot divide by an imaginary number.

  • A fraction should always have a real-number denominator.

• To achieve a real number in the denominator, we need to find a term to multiply numerator and denominator that will eliminate the imaginary part of the denominator.

Complex Conjugate Properties

• The complex conjugate of a + bi is a - bi.

• Reciprocal Relationship:

  • The complex conjugate of a - bi is a + bi.

• Special Property: Product of a complex number and its conjugate yields a real number.

  • Example: (a + bi)(a - bi) = a² + b²

Division Steps of Complex Numbers

• Step 1: Write the complex division as a fraction: ( \frac{c + di}{a + bi} ) where a ≠ 0 and b ≠ 0.

• Step 2: Find the complex conjugate of the denominator.

• Step 3: Multiply both numerator and denominator by the complex conjugate of the denominator: ( \frac{(c + di)(a - bi)}{(a + bi)(a - bi)} ).

• Step 4: Apply distributive property and simplify.

Example of Simplifying Division

• When dividing two complex numbers, substitute values into the steps outlined as necessary.

Working with Complex Conjugates

• When using complex conjugates, results from multiplication and addition yield real numbers:

  • When multiplied together, they simplify to a real number.

  • Example for finding complex conjugates given a number.

Evaluating Functions with Complex Numbers

• General Note: Functions can be evaluated at real or complex numbers, similar to their evaluation at typical scalar numbers.

• Example: function evaluation with complex numbers involves substitution in a polynomial expression.

Simplifying Powers of i

• Cyclic Nature of i: Powers of i repeat every four powers:

  • i¹ = i

  • i² = -1

  • i³ = -i

  • i⁴ = 1

• Observing cyclical patterns is critical when evaluating powers of i.

Exponent Properties with Complex Numbers

• Review exponent properties to simplify powers effectively.