Download 4.3.2 Adding and Subtracting Complex Numbers

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 in.
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4.3.2: Adding and Subtracting Complex Numbers
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
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
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
Guided Practice
Example 2
Is (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
Guided Practice: Example 2, continued
1. 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 • i1 = i
15 ÷ 4 = 3 remainder 3, so 15 = 3 • 4 + 3.
i 15 = i 3 • 4 • i 3 = –i
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 2, continued
Replace 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
Guided Practice: Example 2, continued
2. Distribute the difference through both
parts of the complex number.
(5 + 6i) – [5 + (–3i)]
= (5 + 6i) – 5 – (–3i)
= 5 + 6i – 5 + 3i
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 2, continued
3. Find the sum or difference of the real
parts.
5–5=0
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 2, continued
4. Find the sum or difference of the
imaginary parts.
6i + 3i = 9i
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 2, continued
5. Find the sum of the real and imaginary
parts.
0 + 9i = 9i
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 2, continued
6. 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
Guided Practice: Example 2, continued
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice
Example 3
Is (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
Guided Practice: Example 3, continued
1. 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 = –1
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 3, continued
Replace 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
Guided Practice: Example 3, continued
2. Find the sum or difference of the real
parts.
(12 – 1) + [–18 – (–4)]
= 12 – 1– 18 + 4
= –3
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 3, continued
3. 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
Guided Practice: Example 3, continued
4. Find the sum of the real and imaginary
parts.
–3 + 0i = –3
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4.3.2: Adding and Subtracting Complex Numbers
Guided Practice: Example 3, continued
5. 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
Guided Practice: Example 3, continued
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4.3.2: Adding and Subtracting Complex Numbers