Download 2.1 Complex Numbers Objectives: Perform operations with complex

2.1 Complex Numbers
Objectives: Perform operations with complex
numbers and solve quadratic equations with
complex numbers
Complex Numbers and Imaginary Numbers
The imaginary unit i is defined as
The set of all numbers in the form
a + bi
with real numbers a and b, and i, the imaginary unit,
is called the set of complex numbers.
The standard form of a complex number is
a + bi.
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Operations on Complex Numbers
The form of a complex number a + bi is like the binomial
a + bx. To add, subtract, and multiply complex numbers,
we use the same methods that we use for binomials.
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Example: Adding and Subtracting Complex Numbers
Perform the indicated operations, writing the result
in standard form:
a. (5  2i )  (3  3i )
b. (2  6i )  (12  i )
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Example: Multiplying Complex Numbers
Find the product:
a. 7i (2  9i )
b. (5  4i )(6  7i )
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Conjugate of a Complex Number
For the complex number a + bi, we define its complex
conjugate to be a – bi.
The product of a complex number and its conjugate
is a real number.
(a  bi )(a  bi )
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Complex Number Division
The goal of complex number division is to obtain a real
number in the denominator. We multiply the numerator
and denominator of a complex number quotient by the
conjugate of the denominator to obtain this real number.
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Example: Using Complex Conjugates to Divide Complex
Numbers
Divide and express the result in standard form:
5  4i
4i
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Principal Square Root of a Negative Number
For any positive real number b, the principal square root of
the negative number – b is defined by
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Example: Operations Involving Square Roots of Negative
Numbers
Perform the indicated operations and write the result in
standard form.
b. (2  3) 2
a. 27  48
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Quadratic Equations with Complex Imaginary Solutions
A quadratic equation may be expressed in the general form
ax 2  bx  c  0
and solved using the quadratic formula,
b  b 2  4ac
.
2a
b2 – 4ac is called the discriminant. If the discriminant is
negative, a quadratic equation has no real solutions.
Quadratic equations with negative discriminants have two
solutions that are complex conjugates.
x
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Example: A Quadratic Equation with Imaginary
Solutions
Solve using the quadratic formula: x 2  2 x  2  0
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