4.8 c omplex n umbers part 1: introduction to complex and imaginary numbers
TRANSCRIPT
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4.8 COMPLEX NUMBERSPart 1: Introduction to Complex and Imaginary Numbers
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REAL NUMBERS
See Page 12 in Textbook
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COMPLEX NUMBERS
The set of Real Numbers is a subset of a larger set of numbers called Complex NumbersThe complex numbers are based on
a number whose square root is –1 The imaginary unit i is the complex
number whose square root is –1 .
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SQUARE ROOT OF A NEGATIVE REAL NUMBER
For any real number a,
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EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
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EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
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EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
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REAL AND IMAGINARY NUMBERS
An imaginary number is any number of the form a + bi, where a and b are real numbers and b ≠ 0.
If b = 0, then the number is a real number. If a = 0 and b ≠ 0, then the number is a
pure imaginary number
a + bi↑ ↑
Real Part
ImaginaryPart
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COMPLEX NUMBERS
Imaginary numbers and real numbers make up the set of complex numbers
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POWERS OF IMAGINARY NUMBERS
2
3
4
5
6
7
8
1
1
1
1
1
1
i
i
i i
i
i
i
i i
i
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EVALUATING POWERS
Divide the exponent by 4 and determine the remainder.
Equivalent power depends on the remainder
2
3
4
i
i
i
i
Remainder of 1
Remainder of 2
Remainder of 3
Remainder of 0
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TRY THESE
15
20
201
26
i
i
i
i
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GRAPHING COMPLEX NUMBERS In the complex number plane,
The x – axis represents the real part The y – axis represents the imaginary part
The point (a, b) represents the complex number a + bi
The absolute value of a complex number is its distance from the origin in the complex plane.
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EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?
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EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?
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ADDING AND SUBTRACTING COMPLEX NUMBERS
To add or subtract, combine like terms
3 4 2 5 5 9
3 4 2 5 3 4 2 5 1
i i i
i i i i i
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ADDING AND SUBTRACTING 2
Add or subtract
5 9 25
3 12 2 75
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MULTIPLYING COMPLEX NUMBERS
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THE QUADRATIC FORMULA
Every quadratic equation has complex number solutions (that are sometimes real numbers).
We can use and the quadratic formula to solve all quadratic equations.
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FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION
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FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION
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HOMEWORK
P253 #1, 2, 8 – 17 all, 39 – 44 all