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Introduction Recall that the imaginary unit i is equal to -1. A fraction with i in the denominator does not have a rational denominator, since -1 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 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. c + di ( c + di ) ¸ ( a + bi ) = a + bi 2 4.3.4: Dividing Complex Numbers 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 Key Concepts, continued • In the following equation, let a, b, c, and d be real numbers. c + di ) ( a - bi ) ac + bd + adi - bci ( = = a + bi ( a + bi ) ( a - bi ) a +b c + di 2 2 4 4.3.4: Dividing Complex Numbers 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 Guided Practice Example 2 Find the result of (10 + 6i ) ÷ (2 – i ). 6 4.3.4: Dividing Complex Numbers Guided Practice: Example 2, continued 1. Rewrite the expression as a fraction. (10 + 6i ) ¸ ( 2 - i ) = 10 + 6i 2- i 7 4.3.4: Dividing Complex Numbers 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 Guided Practice: Example 2, continued 3. Rationalize the fraction by multiplying both the numerator and denominator by the complex conjugate of the denominator. 10 + 6i 2-i = = 10 + 6i ) ( 2 + i ) ( = (2 - i )(2 + i ) 20 + 12i + 10i + 6i 2 4 + 2i - 2i - i 2 20 + 22i - 6 4 +1 = 14 + 22i 4.3.4: Dividing Complex Numbers 5 9 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. 14 + 22i 5 = 14 5 + 22 5 i ✔ 10 4.3.4: Dividing Complex Numbers Guided Practice: Example 2, continued 11 4.3.4: Dividing Complex Numbers Guided Practice Example 3 Find the result of (4 – 4i) ÷ (3 – 4i 3). 12 4.3.4: Dividing Complex Numbers Guided Practice: Example 3, continued 1. Simplify any powers of i. i 3 = –i 13 4.3.4: Dividing Complex Numbers 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 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. 4 - 4i ( 4 - 4i ) ¸ ( 3 + 4i ) = 3 + 4i 15 4.3.4: Dividing Complex Numbers 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 Guided Practice: Example 3, continued 5. Rationalize the fraction by multiplying both the numerator and denominator by the complex conjugate of the denominator. 4 - 4i ) ( 3 - 4i ) ( = 3 + 4i ( 3 + 4i ) ( 3 - 4i ) 4 - 4i = = 12 - 12i - 16i + 16i 2 9 + 12i - 12i - 16i 2 12 - 28i - 16 9 - ( -16 ) = -4 - 28i 4.3.4: Dividing Complex Numbers 25 17 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. -4 - 28i 25 =- 4 25 - 28 25 i ✔ 18 4.3.4: Dividing Complex Numbers Guided Practice: Example 3, continued 19 4.3.4: Dividing Complex Numbers