Transcript 5-4 Complex Numbers Day 2

5 - 4:
Complex Numbers
(Day 2)
Objective: CA 5.0:
Students demonstrate knowledge of
how real number and complex numbers
are related both arithmetically and
graphically.
To multiply complex numbers use
the distributive property or the
FOIL method.
Example 5: Write each expression
as a complex number in standard
form.
5i (2  i )
5i (2  i)  10i  5i
2
 10i  5  1
 5  10i
Example 6:
 7  4i  1  2i 
 7  4i  1  2i   7  14i  4i  8i
 7  18i  8  1
 1  18i
2
Example 7:
 6  3i  6  3i 
 6  3i  6  3i   36  18i  18i  9i
 36  9  1
 36  9
 45
2
Complex Conjugates.
Complex conjugates have the form
 a  bi  and  a  bi 
The product of complex conjugates
is always a real number.
(a  bi)(a  bi)  a  abi  abi  b i
2
2 2
 a  b  1
2
2
 a b
2
2
Dividing Complex Numbers
Write the quotient in standard
form.
5  3i
1  2i
The key is to multiply the
numerator and the denominator
by the complex conjugate of the
denominator
5  3i  5  3i   1  2i 



1  2i  1  2i   1  2i 
2
5  10i  3i  6i

2
1  2i  2i  4i
5  13i  6  1

1  4  1
1  13i

5
1 13
 
i
5 5
The absolute value of a
complex number
The absolute value of a complex number z  a  bi,
denoted z , is a nonnegative real number defined as follows
z  a b
2
2
Geometrically, the absolute value
of a complex number is the
numbers distance from the origin in
the complex plane.
Example 8:
Finding the absolute values of a
Complex Number
Which number is the furthest from
the origin in the complex plane?
a 3  4i b.  2i c. 1  5i
a. 3  4i  3  4
2
2
 9  16
 25  5
b. 2i  0  2i
 0   2 
 42
2
c. 1  5i 
 1   5 
2
2
 1  25
 26  5.1
Since  1  5i has the greatest absolute value, it is the
farthest from the origin in the complex plane.
Homework:
Ppage 278 #47 – 71 odd