1. Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational number. Similar to rationalizing a fraction with an irrational square root in the denominator, fractions with i in the denominator can also have the denominator rationalized. 1 4.3.4: Dividing Complex Numbers

2. Key Concepts Any powers of i should be simplified before dividing complex numbers. After simplifying any powers of i, rewrite the division of two complex numbers in the form a + bi as a fraction. To divide two complex numbers of the form a + bi and c + di, where a, b, c and d are real numbers, rewrite the quotient as a fraction. 2 4.3.4: Dividing Complex Numbers

3. Key Concepts, continued Rationalize the denominator of a complex fraction by using multiplication to remove the imaginary unit i from the denominator. The product of a complex number and its conjugate is a real number, which does not contain i. Multiply both the numerator and denominator of the fraction by the complex number in the denominator. Simplify the rationalized fraction to find the result of the division. 3 4.3.4: Dividing Complex Numbers

4. Key Concepts, continued In the following equation, let a, b, c, and d be real numbers. 4 4.3.4: Dividing Complex Numbers

5. Common Errors/Misconceptions multiplying only the denominator by the complex conjugate incorrectly determining the complex conjugate of the denominator 5 4.3.4: Dividing Complex Numbers

6. Guided Practice Example 2 Find the result of (10 + 6i ) ÷ (2 – i ). 6 4.3.4: Dividing Complex Numbers

7. Guided Practice: Example 2, continued 1.Rewrite the expression as a fraction. 7 4.3.4: Dividing Complex Numbers

8. Guided Practice: Example 2, continued 2.Find the complex conjugate of the denominator. The complex conjugate of a – bi is a + bi, so the complex conjugate of 2 – i is 2 + i. 8 4.3.4: Dividing Complex Numbers

9. Guided Practice: Example 2, continued 3.Rationalize the fraction by multiplying both the numerator and denominator by the complex conjugate of the denominator. 9 4.3.4: Dividing Complex Numbers

10. Guided Practice: Example 2, continued 4.If possible, simplify the fraction. The answer can be left as a fraction, or simplified by dividing both terms in the numerator by the quantity in the denominator. 10 4.3.4: Dividing Complex Numbers

11. Guided Practice: Example 2, continued 11 4.3.4: Dividing Complex Numbers

12. Guided Practice Example 3 Find the result of (4 – 4i) ÷ (3 – 4i 3 ). 12 4.3.4: Dividing Complex Numbers

13. Guided Practice: Example 3, continued 1.Simplify any powers of i. i 3 = –i 13 4.3.4: Dividing Complex Numbers

14. Guided Practice: Example 3, continued 2.Simplify any expressions containing a power of i. 3 – 4i 3 = 3 – 4(–i) = 3 + 4i 14 4.3.4: Dividing Complex Numbers

15. Guided Practice: Example 3, continued 3.Rewrite the expression as a fraction, using the simplified expression. Both numbers should be in the form a + bi. 15 4.3.4: Dividing Complex Numbers

16. Guided Practice: Example 3, continued 4.Find the complex conjugate of the denominator. The complex conjugate of a + bi is a – bi, so the complex conjugate of 3 + 4i is 3 – 4i. 16 4.3.4: Dividing Complex Numbers

17. Guided Practice: Example 3, continued 5.Rationalize the fraction by multiplying both the numerator and denominator by the complex conjugate of the denominator. 17 4.3.4: Dividing Complex Numbers

18. Guided Practice: Example 3, continued 6.If possible, simplify the fraction. The answer can be left as a fraction, or simplified by dividing both terms in the numerator by the quantity in the denominator. 18 4.3.4: Dividing Complex Numbers

19. Guided Practice: Example 3, continued 19 4.3.4: Dividing Complex Numbers