Transcript More with Complex Numbers - Northland Preparatory Academy

Sec. 2.5b
More with
Complex Numbers
Definition: Complex Conjugate
The complex conjugate of the complex number
z  a  bi
is
z  a  bi  a  bi
What happens when we multiply a complex number by
its conjugate???
 a  bi   a  bi   a
2
 abi  abi   bi 
 a b
2
2
2 This is a positive
real number!!!
Practice Problems
Write the given complex numbers in standard form.
2 3  i 6  2i 6 2
3 1
 2 2  i  i
3  i 3  i 3  1 10 10
5 5
5  i 2  3i 10  15i  2i  3i
7  17i


2
2
2  3i 2  3i
13
2 3
7 17
  i
13 13
2
Complex Solutions of Quadratic
Equations
Remind me of the quadratic formula!!!
b  b  4ac
x
2a
2
What’s this
part called?
 The discriminant!!!
It can be used to tell whether the solutions to a particular
quadratic equation are real numbers…
Discriminant of a Quadratic
Equation
ax  bx  c  0, where a, b, and c
are real numbers and a  0 ,
For a quadratic equation
• If
• If
2
b  4ac  0 , there are two distinct real solutions.
2
b  4ac  0 , there is one repeated real solution.
2
b  4ac  0 , there is a complex conjugate pair
2
• If
of solutions.
Practice Problems
Solve
x  x 1  0
2
a = b = c = 1  Use the quadratic formula!
x
 1 
1  4 11
2 1
2
1  3

2
A complex
conjugate pair
1
3
 
i
2 2
Guided Practice
Write the given complex number in standard form.
1  i   1  2i  i  1  i    2i 1 i 
3
2
 2i  2i  2  2i
2
Guided Practice
Write the given expression in standard form.
2  i 3i 6i  3i
1 2
  i

3i 3i
3 3
9
2
Guided Practice
Write the given expression in standard form.
 2  i 1  2i 
5  2i


2  4i  i  2i  5  2i 
5  2i

2
2
5  2i
5  4i
2
4  3i  5  2i  20  8i  15i  6i 2



29
29
26 7

 i
29 29
The Complex Plane
Imaginary Axis
bi
Imaginary Axis
a  bi
a
2  3i
3i
Real Axis
2
Real Axis
The Complex Plane
Plot u = 1 + 3i, v = 2 – i, and u + v in the complex plane.
Imaginary Axis
u  1  3i
u  v  3  2i
Notice that the two
complex numbers, their
sum, and the origin form
a quadrilateral (what type?)
Real Axis
v  2i
 A Parallelogram!!!
Definition: Absolute Value of a Complex Number
The absolute value, or modulus, of the complex number
z  a  bi , where a and b are real numbers, is
z  a  bi  a  b
2
2
Imaginary Axis
z  a  bi
bi
z
a
Real Axis
A Few More New Formulas
The distance between the points u and v in the
complex plane:
d  u v
The midpoint of the line segment connecting u and v
in the complex plane:
uv
2
A Few More New Formulas
Find the distance between u = –4 + i and v = 2 + 5i in the
complex plane, and find the midpoint of the segment
connecting u and v.
Distance:
u  v   4  i    2  5i   6  4i

Midpoint:
 6   4  2 13  7.211
2
2
Can we verify these
answers graphically?
u  v 2  6i
 1  3i

2
2
Whiteboard Problems…
Write the given complex number in standard form.
3
3
 3 1  1
3 i
 i    

 2 2  2
1
1
2
 3  2 3i  i
3  i  1  3i
8
4
1
1
2

3  i  3i  3i   4i   i
4
4


3







3i
