Transcript Section 2.4 Complex Numbers

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
Natural
Whole
{0, 1, 2, 3, 4,…}
How many whole
numbers are
there?
Real Number System
Natural
Whole
Integers
{...-3, -2, -1, 0, 1, 2, 3, …}
How many integers
numbers are there?
Real Number System
Natural
Whole
Integers
Rational
Fractions
How many rational
numbers are
there?
a

 a, b  I , b  0
b

Real Number System
Natural
Whole
Integers
Rational
2,  , e
How many
irrational numbers
are there?
Irrational
Real Number System
Natural
Whole
Integers
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
Rational
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
The number i is such that

1  i
2
2
1  i

1  i
2
i  1
Imaginary Unit
b  i b
16  i 16  4i
Complex Numbers
REAL
a  bi
Imaginary
Complex
3  2i
3 2
 i
8 5
7  0i
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  4i
  81   9i
7  i 7
If you square a radical you get
the radicand
 5
2
5
Whenever you have
i2 the next turn you
will have -1 and no
i.
i  
2
1
i  1
2

2
Equality of Complex numbers
If a + bi = c + di, then a = c and b = d.
x  5i  7  yi
x7
y 5
Is a negative times a negative
always positive?
Trick question. This is not a negative times a negative.
 9   25 
(3i)(5i) 15i   15
2
Example
 7   7  i 7 i 7
 7i
2
 7
Example
 5   10  i 5 i 2  5
 5i
2
2
 5 2
Example
 15  2  i 15  2
 i 30
Example
 32
i 32

2
i 2
 16
4
Cancel the
i
factor
Add
Collect like terms.
(3  5i)  (4  7i)
7  2i
First distribute the
negative sign.
Subtract
Now collect like
terms.
(5  7i)  (4  20i)
 5  7i  4  20i
 9  13i
Multiplication
(3  2i)(4  5i)
2
12  15i  8i  10i
12  7i  10
22  7i
FO I L
Simplify each expression. Express
your answer in form.
(5  4i)(3  7i) 
F-O-I-L
15  35i  12i  28i
2
Recall
i2=-1
Combine like terms.
 15  23i  28  43 23i
Combine like terms.
Write in the form
a  bi.
26  3  2i  26(3  2i )

 
2
3  2i  3  2i 
9  4i
2 26(3  2i )

13
 2(3  2i)  6  4i
Multiply by the
conjugate factor.
Powers of
i
1  i raised to the 0 is 1.
1 Anything raised to the 1 is itself.
i i
2
2
i  1
1  i
3
3
2
 i  i i  i i  (1)i i
0 Anything other than 0
Simplify as much as possible.
i  i i  (1)(1)  1
4
2 2
i  (i )  i  (1)(1)  1
30
4 7
2
Use the Quadratic Formula
2
2
9 x  6 x  37  0
 b  b  4ac
x
a 9
2a
b  6
 (6)  (6) 2  4(9)(37)
x

c  37
2(9)
6  36i
6  36  1332 6   1296



18
18
18
6 36
1
  i   2i
18 18
3
Homework Section 2.4
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