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Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic quantities. The real parts of the complex number are similar to the numeric quantities, and the imaginary parts of the complex number are similar to the algebraic quantities. Before finding sums or differences, each complex number should be in the form a + bi. If i is raised to a power n, use the remainder of n ÷ 4 to simplify i n . 1 4.3.2: Adding and Subtracting Complex Numbers

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Page 1: Introduction

IntroductionFinding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic quantities. The real parts of the complex number are similar to the numeric quantities, and the imaginary parts of the complex number are similar to the algebraic quantities. Before finding sums or differences, each complex number should be in the form a + bi. If i is raised to a power n, use the remainder of n ÷ 4 to simplify in.

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4.3.2: Adding and Subtracting Complex Numbers

Page 2: Introduction

Key Concepts• First, find the sum or difference of the real parts of the

complex number. • Then, to find the sum or difference of the imaginary

numbers, add or subtract the coefficients of i. • The resulting sum of the real parts and the imaginary

parts is the solution.• In the following equation, let a, b, c, and d be real

numbers. (a + bi) + (c + di) = a + c + bi + di = (a + c) + (b + d)i

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4.3.2: Adding and Subtracting Complex Numbers

Page 3: Introduction

Key Concepts, continued• a + c is the real part of the sum, and (b + d)i is the

imaginary part of the sum. • When finding the difference, distribute the negative

throughout both parts of the second complex number.(a + bi) – (c + di) = a + bi – c – di = (a – c) + (b – d)i

• a – c is the real part of the difference, and (b – d)i is the imaginary part of the difference.

• The sum or difference of two complex numbers can be wholly real (having only real parts), wholly imaginary (having only imaginary parts), or complex (having both real and imaginary parts).

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4.3.2: Adding and Subtracting Complex Numbers

Page 4: Introduction

Common Errors/Misconceptions• failing to distribute a negative throughout both the real

and imaginary parts of a complex number before simplifying a difference

• adding the multiples of two powers of i when the powers of i are not equal, such as 2i 2 + 3i = 5i

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4.3.2: Adding and Subtracting Complex Numbers

Page 5: Introduction

Guided PracticeExample 2Is (5 + 6i 9) – (5 + 3i15) wholly real or wholly imaginary, or does it have both a real and an imaginary part?

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4.3.2: Adding and Subtracting Complex Numbers

Page 6: Introduction

Guided Practice: Example 2, continued1. Simplify any expressions containing in.

Two expressions, 6i 9 and 3i15, contain in. Divide each power of i by 4 and use the remainder to simplify in. 9 ÷ 4 = 2 remainder 1, so 9 = 2 • 4 + 1.

i 9 = i 2 • 4 • i 1 = i 15 ÷ 4 = 3 remainder 3, so 15 = 3 • 4 + 3.

i 15 = i 3 • 4 • i 3 = –i6

4.3.2: Adding and Subtracting Complex Numbers

Page 7: Introduction

Guided Practice: Example 2, continuedReplace each occurrence of in in the expressions with the simplified versions, and replace the original expressions in the difference with the simplified expressions.

6i 9 = 6 • (i) = 6i 3i 15 = 3 • (–i) = –3i (5 + 6i 9) – (5 + 3i 15) = (5 + 6i) – [5 + (–3i)]

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4.3.2: Adding and Subtracting Complex Numbers

Page 8: Introduction

Guided Practice: Example 2, continued2. Distribute the difference through both

parts of the complex number.

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4.3.2: Adding and Subtracting Complex Numbers

Page 9: Introduction

Guided Practice: Example 2, continued3. Find the sum or difference of the real

parts. 5 – 5 = 0

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4.3.2: Adding and Subtracting Complex Numbers

Page 10: Introduction

Guided Practice: Example 2, continued4. Find the sum or difference of the

imaginary parts. 6i + 3i = 9i

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4.3.2: Adding and Subtracting Complex Numbers

Page 11: Introduction

Guided Practice: Example 2, continued5. Find the sum of the real and imaginary

parts. 0 + 9i = 9i

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4.3.2: Adding and Subtracting Complex Numbers

Page 12: Introduction

Guided Practice: Example 2, continued6. Use the form of the sum to determine if it

is wholly real or wholly imaginary, or if it has both a real and an imaginary part.

9i has only an imaginary part, 9i, so the difference is wholly imaginary.

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4.3.2: Adding and Subtracting Complex Numbers

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Page 13: Introduction

Guided Practice: Example 2, continued

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4.3.2: Adding and Subtracting Complex Numbers

http://www.walch.com/ei/00225
Page 14: Introduction

Guided PracticeExample 3Is (12 – I 20) + (–18 – 4i 18) wholly real or wholly imaginary, or does it have both a real and an imaginary part?

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4.3.2: Adding and Subtracting Complex Numbers

Page 15: Introduction

Guided Practice: Example 3, continued1. Simplify any expressions containing in.

Two expressions, i 20 and 4i 18, contain in. Divide each power of i by 4 and use the remainder to simplify in. 20 ÷ 4 = 5 remainder 0, so 20 = 5 • 4 + 0.

i 20 = i 5 • 4 = 1 18 ÷ 4 = 4 remainder 2, so 18 = 4 • 4 + 2.

i 18 = i 4 • 4 • i 2 = –115

4.3.2: Adding and Subtracting Complex Numbers

Page 16: Introduction

Guided Practice: Example 3, continuedReplace each occurrence of in in the expressions with the simplified versions, and replace the original expressions in the difference with the simplified expressions.

i 20 = 1 4i 18 = 4 • (–1) = –4 (12 – i 20) + (–18 – 4i 18) = (12 – 1) + [–18 – (–4)]

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4.3.2: Adding and Subtracting Complex Numbers

Page 17: Introduction

Guided Practice: Example 3, continued2. Find the sum or difference of the real

parts.

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4.3.2: Adding and Subtracting Complex Numbers

Page 18: Introduction

Guided Practice: Example 3, continued3. Find the sum or difference of the

imaginary parts. The expression contains no multiples of i, so there are no imaginary parts, and the multiple of i is 0: 0i.

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4.3.2: Adding and Subtracting Complex Numbers

Page 19: Introduction

Guided Practice: Example 3, continued4. Find the sum of the real and imaginary

parts. –3 + 0i = –3

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4.3.2: Adding and Subtracting Complex Numbers

Page 20: Introduction

Guided Practice: Example 3, continued5. Use the form of the sum to determine if it

is wholly real or wholly imaginary, or if it has both a real and an imaginary part.

–3 has only a real part, –3, so the sum is wholly real.

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4.3.2: Adding and Subtracting Complex Numbers

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Page 21: Introduction

Guided Practice: Example 3, continued

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4.3.2: Adding and Subtracting Complex Numbers

http://www.walch.com/ei/00226

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