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Drill #81: Solve each equation or inequality
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Drill #83: Simplify each expression
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5-9 Complex Numbers Objective: To simplify square roots containing negative numbers and to add, subtract, and multiply complex numbers
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(1.) i Definition: i is called the imaginary unit.
What is the value of I squared?
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(2.) Pure Imaginary Numbers
Definition: For any positive number b, where i is the imaginary unit, and bi is called a pure imaginary number. Example:
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Evaluating the Square Root of Negative Numbers*
To find the square root of negative numbers: 1. First separate the negative 2. Evaluate each root separately and multiply
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More Examples Ex1. Ex2.
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Powers of I *
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Finding Powers of i* Powers of i are cyclical. They repeat after
To find : 1. Divide n by 4 and keep only the remainder r 2. , where r is the remainder of n/4 Note:
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Example (Powers of i) Find the following: Ex Ex2. Ex3.
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(3.) Complex Numbers Definition: A number in the form of a + bi
where a and b are real numbers and i is the imaginary unit. a is called the real part. b is called the imaginary part.
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Adding Complex Numbers*
To add complex numbers: 1. add the real parts together (this is the real part of sum) 2. add imaginary parts together (this is the imaginary part of the solution). (a + bi) + (c + di) = (a + c) + (b + d)i Ex: (5 + 6i) + (2 + 3i) = 7 + 9i
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Adding Complex Numbers
Examples: Ex1. (2 + 9i) + (3 + 4i) Ex2. (5 + 6i) - (2 + 3i) Ex3.
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Multiplying Complex Numbers*
Definition: To multiply imaginary numbers you need to FOIL. (a + bi)(c + di) = = ac (first) + adi (outside) + bci (inside) + bd (last) = (ac - bd) + (ad + bd)i Ex: (2 + 3i)(4 + 5i) = 2(4) + 2(5i) + 3i(4) + 3i(5i) = i + 12i – 15 = i
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Multiplying Complex Numbers
(2 + 3i)( 1 + 4i) Ex2. (6 + 2i)( 3 – 2i) Ex3. (3 – 5i)(3 + 5i)
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(4.) Complex Conjugates Definition: Numbers of the form
a + bi and a – bi are called complex conjugates. The product of complex conjugates is: Example: 3 + 2i and 3 – 2i are complex conj.
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The Product of Complex Conjugates*
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Dividing by Complex Numbers* Rationalizing Complex Denominators
To rationalize a complex denominator you need to multiply the numerator and denominator by the complex conjugate. Example: Simplify
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Solving 2nd Equations* (no 1st degree term)
To solve equations: 1. Isolate the square term. 2. Take the (+/-) square root of both sides. Example:
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(5.) Equal Complex Numbers
Definition: a + bi = c + di if and only if a = c and b = d. The real parts must be equal and the imaginary parts must be equal.
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Equal Complex Numbers*
Find values of x and y for which each equation is true:
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