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Section 2.4 Complex Numbers
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 find complex solutions to quadratic equations
Real Number System
Natural {1, 2, 3, 4,…}
How many
natural numbers
are there?
Real Number System
Whole
Natural {0, 1, 2, 3, 4,…}
How many whole
numbers are
there?
Real Number System
Integers
Whole
Natural {...-3, -2, -1, 0, 1, 2, 3, …}
How many integers numbers are there?
Real Number System
Rational
Integers
Whole
Natural Fractions
How many rational numbers are there?
0,, bIba
b
a
Real Number System
Rational
Integers
Whole
Natural
How many irrational numbers are there?
e,,2
Irrational
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
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
What square root of -16?
Whatever it is it is not on the real number line.
Definition of i
b bi1 i
The number i is such that 1i
221 i
21 i 16 16i 4i
Imaginary Unit
ImaginaryREAL
ComplexComplex
Complex Numbers
bia
i23 i5
2
8
3 i07
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.
Examples
16
81
7
i4
i9
7i
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.
Equality of Complex numbers
If a + bi = c + di, then a = c and b = d.
yiix 75
7x 5y
Is a negative times a negative always positive?
259
)5)(3( ii 215i 15
Trick question. This is not a negative times a negative.
Example
77 77 ii 27i
7
Example
105 525 ii
25 2i
25
Example
215 215 i
30i
Example
2
32
2
32
i
i
16
4
Cancel the i
factor
Add
)74()53( ii
7 i2
Collect like terms.
Subtract
9 i13
)204()75( ii
ii 20475
First distribute the negative sign.
Now collect like terms.
Multiplication
)23( i )54( i
F O I L
i1512 i8 210i10712 i
i722
Simplify each expression. Express your answer in form.
)73)(45( ii228123515 iii
Combine like terms.
282315 i i2343
Recall i2=-1
F-O-I-L
Combine like terms.
Write in the form
i23
26
.bia
i
i
23
23249
)23(26
i
i
13
)23(26 i )23(2 i
Multiply by the conjugate factor.
2i46
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
Simplify as much as possible.
4i 2 2i i ( 1)( 1) 1
30i 4 7 2( )i i (1)( 1) 1
Use the Quadratic Formula
03769 2 xx9a
6b37c
a
acbbx
2
42
)9(2
)37)(9(4)6()6( 2 x
18
1332366
18
12966
18
366 i
i18
36
18
6 i2
3
1
Homework Section 2.41-79, 83 odd