Download 4.3.3 Multiplying Complex Numbers

Introduction
The product of two complex numbers is found using the
same method for multiplying two binomials. As when
multiplying binomials, both terms in the first complex
number need to be multiplied by both terms in the
second complex number. The product of the two
binomials x + y and x – y is the difference of squares:
x2 – y2. If y is an imaginary number, this difference of
squares will be a real number since i • i = –1:
(a + bi)(a – bi) = a2 – (bi)2 = a2 – b2(–1) = a2 + b2.
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4.3.3: Multiplying Complex Numbers
Key Concepts
• Simplify any powers of i before evaluating products of
complex numbers.
• In the following equations, let a, b, c, and d be real
numbers.
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4.3.3: Multiplying Complex Numbers
Key Concepts, continued
• Find the product of the first terms, outside terms,
inside terms, and last terms. Note: The imaginary unit
i follows the product of real numbers.
(a + bi ) • (c + di ) = ac (product of the first terms) +
adi (product of the outside terms) + bci (product of
the inside terms) + bidi (product of the last terms)
= ac + adi + bci + bdi2
= ac + bd(–1) + adi + bci
= (ac – bd) + (ad + bc)i
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4.3.3: Multiplying Complex Numbers
Key Concepts, continued
• ac – bd is the real part of the product, and ad + bc is
the multiple of the imaginary unit i in the imaginary
part of the product.
• A complex conjugate is a complex number that
when multiplied by another complex number produces
a value that is wholly real.
• The product of a complex number and its conjugate is
a real number.
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4.3.3: Multiplying Complex Numbers
Key Concepts, continued
• The complex conjugate of a + bi is a – bi, and the
complex conjugate of a – bi is a + bi.
• The product of a complex number and its conjugate is
the difference of squares, a2 – (bi)2, which can be
simplified.
a2 – b 2i 2 = a2 – b2 • (–1) = a2 + b2
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4.3.3: Multiplying Complex Numbers
Common Errors/Misconceptions
• incorrectly finding the product of two complex numbers
• incorrectly identifying the complex conjugate of a + bi
as a value such as –a + bi, a + bi, or –a – bi
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4.3.3: Multiplying Complex Numbers
Guided Practice
Example 1
Find the result of i 2 • 5i.
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 1, continued
1. Simplify any powers of i.
i 2 · i = i 3 = -i
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 1, continued
2. Multiply the two terms. Simplify the
expression, if possible, by simplifying any
remaining powers of i or combining like
terms.
5(–i) = –5i
✔
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 1, continued
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4.3.3: Multiplying Complex Numbers
Guided Practice
Example 2
Find the result of (7 + 2i)(4 + 3i).
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 2, continued
1. Multiply both terms in the first polynomial
by both terms in the second polynomial.
Find the product of the first terms,
outside terms, inside terms, and last
terms.
(7 + 2i)(4 + 3i) = 7 • 4 + 7 • 3i + 2i • 4 + 2i • 3i
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 2, continued
2. Evaluate or simplify each expression.
7 • 4 + 7 • 3i + 2i • 4 + 2i • 3i
= 28 + 21i + 8i + 6i 2
= 28 + 21i + 8i + 6(-1)
= 28 + 21i + 8i - 6
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 2, continued
3. Combine any real parts and any imaginary
parts.
28 + 21i + 8i - 6
= 28 - 6 + 21i + 8i
= 22 + 29i
✔
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4.3.3: Multiplying Complex Numbers
Guided Practice: Example 2, continued
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4.3.3: Multiplying Complex Numbers