introduction recall that the imaginary unit i is equal to. a fraction with i in the denominator does...

IntroductionRecall 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.

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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.

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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.

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4.3.4: Dividing Complex Numbers

Key Concepts, continued• In the following equation, let a, b, c, and d be real

numbers.

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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

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4.3.4: Dividing Complex Numbers

Guided Practice

Example 2Find the result of (10 + 6i ) ÷ (2 – i ).

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4.3.4: Dividing Complex Numbers

Guided Practice: Example 2, continued

1. Rewrite the expression as a fraction.

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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.

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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.

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4.3.4: Dividing Complex Numbers

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.

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4.3.4: Dividing Complex Numbers

✔

Guided Practice: Example 2, continued

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4.3.4: Dividing Complex Numbers

Guided Practice

Example 3Find the result of (4 – 4i) ÷ (3 – 4i

3).

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4.3.4: Dividing Complex Numbers

Guided Practice: Example 3, continued

1. Simplify any powers of i. i 3 = –i

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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

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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.

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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.

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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.

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4.3.4: Dividing Complex Numbers

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.

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4.3.4: Dividing Complex Numbers

✔

Guided Practice: Example 3, continued

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4.3.4: Dividing Complex Numbers