Download - Section 1.5 Complex Numbers
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Section 1.5 Complex Numbers
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What you should learn• How to use the imaginary unit i to write
complex numbers• How to add, subtract, and multiply complex
numbers• How to use complex conjugates to write the
quotient of two complex numbers in standard form
• How to use the Quadratic Formula to find complex solutions to quadratic equations.
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Real Number System
Rational
Integers
Whole
Natural
How many irrational numbers are there?
e,,2
Irrational
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Real Number System
Rational
Integers
Whole
Natural Each set is a subset of the Real Number System.
The union of all these sets forms the real number system.
The number line is our model for the real number system.
Irrational
Real
Numbers
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Definition of Square Root
If a2 = n then a is a square root of n.42 = (4)(4) = 16
4 is a square root of 16(-4)2 = (-4)(-4) = 16
-4 is a square root of 16
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What is the square root of -16?
Whatever it is it is not on the real number line.
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Definition of i
b bi1 i
The number i is such that 1i
221 i
21 i 16 16i 4i
Imaginary Unit
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ImaginaryREAL
Complex
Complex Numbers
bia
i23 i52
83 i07
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Definition of a Complex Number• If a and b are real numbers, the number a + bi
is a complex number, and it is said to be written in standard form.
• If b = 0 then the number a + bi = a is a real number.
• If b ≠ 0, then the number a + bi is called an imaginary number.
• A number of the form bi, where b ≠ 0 is called a pure imaginary number.
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Examples
16
81
7
i4
i9
7i
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If you square a radical you get the radicand
1 i 25 5
12i
2 2
Whenever you have i2 the next turn you will have -1 and no
i.
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Equality of Complex numbers
If a + bi = c + di, then a = c and b = d.
yiix 75
7x 5y
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Is a negative times a negative always positive?
259
)5)(3( ii 215i 15
Trick question. This is not a negative times a negative.
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Example
77 77 ii 27i
7
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Example
105 525 ii
25 2i
25
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Example
215 215 i30i
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Example
232
232
ii
16
4
Cancel the i
factor
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Add
)74()53( ii
7 i2
Collect like terms.
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Subtract
9 i13
)204()75( ii
ii 20475
First distribute the negative sign. Now collect like
terms.
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Multiplication
)23( i )54( i
F O I L
i1512 i8 210i10712 ii722
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Simplify each expression. Express your answer in a + bi form.
)73)(45( ii228123515 iii
Combine like terms.
282315 i i2343
Recall i2=-1
F-O-I-L
Combine like terms.
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Write in the form
i2326
.bia
ii
2323
249)23(26
ii
13)23(26 i
)23(2 i
Multiply by the conjugate factor.
2i46
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Powers of i0i11i2i3i
Anything other than 0 raised to the 0 is 1.
Anything raised to the 1 is itself.i12 i1
iii 23 i)1( ii
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Simplify as much as possible.
4i 2 2i i ( 1)( 1) 1
30i 4 7 2( )i i (1)( 1) 1
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Homework Section 1.5 Complex Numbers Page 129
1-4, 17-39 odd, 49-55 odd, 63, 65, 73, 88