Download Conjugate of a Complex Number

Avon High School
Section: 1.4
ACE COLLEGE ALGEBRA II - NOTES
Complex Numbers
Mr. Record: Room ALC-129
Day 1 of 1
The Imaginary Unit i
The Imaginary Unit i
The imaginary unit i is defined as
i  1, where i 2  1.
Equality of Complex Numbers
a  bi  c  di if and only if a  c and b  d
Complex Numbers
a  bi
For the complex number, 4  6i
a, the real
part is -4
Example 1
a. (2  6i )  (12  i)
b, the imaginary
part is 6
Real Numbers
a  bi with b  0
Operations with Complex Numbers
Perform the given operation and simplify.
Conjugate of a Complex Number
The complex conjugate of the number a  bi is a  bi and vice versa.
The multiplication of two complex conjugates gives a real number.
 a  bi  a  bi   a 2 +b 2
 a  bi  a  bi   a 2 +b 2
Imaginary Numbers
a  bi with b  0
b. (7  3i)(2  5i)
Example 2
a.
Using Complex Conjugates to Divide Complex Numbers
Divide and express the result in standard form.
5  4i
4i
b.
7  4i
2  5i
Roots of Negative Numbers
The square root of 4i and the square root of 4i both result in 16 :
(4i)2  16i 2  16(1)  16
(4i)2  16i 2  16(1)  16
Consequently, in the complex number system 16 has two square roots, namely 4i and 4i .
We call 4i the principal square root of 16
Principal Square of a Negative Number
For any positive real number b, the principal square root of the negative number b is defined by
b  i b
Example 4
a.
27  48
Operations Involving Square Roots of Negative Numbers
Perform the indicated operations and write the result in standard form.
b.
 2 
3

2
c.
14  12
2