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Transcript
```4.2 Class notes. Complex Numbers
Some quadratics have no real solutions.
To continue solving, use imaginary units.
i
i5 
i2 
i6 
i3 
i 43 
i4 
Rewrite using imaginary numbers:
1.
16
2.
81
3.
 36
4.
10
5.
18
6.
 28
Solve:
2 x2  11  37
1.
2.
2 x2  18  72
A complex number written in standard form is a number
a  bi where a and b are real numbers.
a is the real part and bi is the imaginary part.


To add/subtract complex numbers, add/subtract their real parts and their imaginary parts
separately.
To multiply complex numbers, use the distributive property or FOIL as normal.
Simplify:
1.
 3  i    2  3i 
2.
2i   4  2i 
3.
3   2  3i    5  i 
4.
 6  2i    5  4i 
5.
i   3i
6.
i  4  3i 
7.
3i   5i
9.
 2  i  4  3i 
8.
10.
a  bi
Two complex numbers of the form
and
of complex conjugates is always a real number.
6i  4  2i 
 6  2i 
2
a  bi are called complex conjugates.
The product
Write in standard form:
1.
7  5i
1  4i
2.
5
1 i
Graphing Complex Numbers:
 Horizontal Axis = Real Axis
 Vertical Axis = Imaginary Axis
Plot the complex numbers:
1.
3  2i
2.
2  4i
3.
3i
4.
4  3i
imaginary
real
Voltage Applications.
Voltage= I* Z, where I – is the current and Z- total impedance
Find the total impedance and voltage for each problem is the circuit has a current of 3-2i.
a) Z=
V=
Hopefully, this will help to answer the question "Does anyone ever really use complex numbers?"
```