4.8 c omplex n umbers part 1: introduction to complex and imaginary numbers
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4.8 COMPLEX NUMBERSPart 1: Introduction to Complex and Imaginary Numbers
REAL NUMBERS
See Page 12 in Textbook
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 .
SQUARE ROOT OF A NEGATIVE REAL NUMBER
For any real number a,
EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
EXAMPLE: SIMPLIFY EACH NUMBER BY USING THE IMAGINARY NUMBER
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
COMPLEX NUMBERS
Imaginary numbers and real numbers make up the set of complex numbers
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
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
TRY THESE
15
20
201
26
i
i
i
i
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.
EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?
EXAMPLE: WHAT ARE THE GRAPH AND ABSOLUTE VALUE OF EACH NUMBER?
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
ADDING AND SUBTRACTING 2
Add or subtract
5 9 25
3 12 2 75
MULTIPLYING COMPLEX NUMBERS
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.
FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION
FIND ALL SOLUTIONS TO EACH QUADRATIC EQUATION
HOMEWORK
P253 #1, 2, 8 – 17 all, 39 – 44 all