Download 4-8 Complex Numbers

Find complex number solutions by
applying rules for graphing and
performing complex number operations.
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𝑖 = −1
So, 𝑖 2 = −1
𝑖 3 = 𝑖 2 ∙ 𝑖 = −𝑖
𝑖 4 = 𝑖 2 ∙ 𝑖 2 = −1 ∙ −1 = 1
Keep this in mind for simplifying imaginary numbers.
• Imaginary numbers exist wherever you have the
square root of a negative number.
•
−𝑎 = −1 ∙ 𝑎 = −1 ∙ 𝑎 = 𝑖 𝑎
•
−18
•
−1 ∙ 2 ∙ 3 ∙ 3
• 3𝑖 2
• You try: −12
• 2𝑖 3
• So is it true that −64 = − 64?
• No!
• 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. (ex: 6+0i = 6)
• If a = 0 and b ≠ 0 then the number is a pure imaginary
number. (ex: 0 + 3i = 3i)
• Imaginary numbers and real numbers make up the set
of complex numbers.
• The x-axis represents the real axis
• The y-axis represents the imaginary axis
• A complex number is of the form a+bi
• a is the real part and b is the imaginary part
• Ex: graph the number 3 − 2𝑖
• This is represented by the point (a, b) or (3, -2)
• The absolute value of a complex number is its distance
from the origin.
• 𝑎 + 𝑏𝑖 = 𝑎2 + 𝑏 2
• Think Pythagorean Theorem
• Find the graph and absolute value of each:
• -5 + 3i
• Graph the point (-5,3) and the absolute value is 34
• 6i
• Graph the point (0,6) and the absolute value is 6
• To add and subtract, combine like terms.
• If you are subtracting, distribute the negative first
• Ex: (5 – 3i) – (-2+4i) = 7-7i
• To multiply, you distribute.
• Use FOIL (First Outer Inner Last) if both numbers have two
terms.
• Ex: 4 + 3𝑖 −1– 2𝑖 = −4 − 8𝑖 − 3𝑖 − 6𝑖 2
• Remember that 𝑖 2 = −1 so we have 2 − 11𝑖
• Odds p.253 #9-25
• Complex Conjugates are the number pairs 𝑎 + 𝑏𝑖
and 𝑎 − 𝑏𝑖
• The product of complex conjugates is always a real number
𝑎2 + 𝑏 2
• To divide complex numbers, you must multiply by the
conjugate of the denominator.
•
9+12𝑖
Ex:
3𝑖
• 4 − 3𝑖
•
2+3𝑖
You try:
1−4𝑖
10
11
•− + 𝑖
17
17
needs to be multiplied by
−3𝑖
−3𝑖
• Factor 2𝑥 2 + 32
• Factor out the GCF: 2(𝑥 2 + 16)
• We know how to factor a difference of squares, but
now we have sum of squares. This means the factors
are the conjugates (x + 4i) and (x – 4i)
• So we have 2(𝑥 + 4𝑖)(𝑥 − 4𝑖)
• Find the solutions to 2𝑥 2 − 3𝑥 + 5 = 0
• Easiest to use quadratic formula.
• 𝑥=
−(−3)± (−3)2 −4(2)(5)
2(2)
• 𝑥=
3± 9−40
4
• 𝑥=
3± −31
4
3
4
• 𝑥= ±
31
𝑖
4
• Odds p.253 #27-53