Transcript 1-10 Introduction to Complex Numbers

1-10 Introduction to Complex
Numbers
What is a complex number?
To see a complex number we have
to first see where it shows up
Solve both of these
x  81  0
2
x  81
x  9
2
x  81  0
2
x  81
2
Uhoh…….what do I do
here?
Um, no solution????
x   81 does not have a real answer.
It has an “imaginary” answer.
To define a complex number we have to
create a new variable.
This new variable is “ i “
Definition: i  1
Note: i is the representation for 1 , not a
simplification of 1
So, following this definition:
i  1
2
i  i
3
i 1
4
And it cycles….
i  1
i5  i4  i  i
i  i i  i
i  1
i  i  i  1
i  i  i  1
i  i
i  i  i  i
i  i  i  i
i 1
i  1 i  1
i  1 i  1
2
3
4
6
7
8
4
4
4
2
3
4
9
10
11
12
Do you see a pattern yet?
8
8
8
8
2
3
4
What is that pattern?
We are looking at the remainder when the
power is divided by 4.
Why?
Every i 4 doesn’t matter. It is what remains
4
after all of the i are taken out.
Try it with i
92233
Hints to deal with i
1. Find all “i”s at the beginning of a problem.
2. Treat all “i”s like variables, with all rules of
exponents holding.
3. Reduce the power of i at the end by the
rules we just learned..
Examples
1.
36  81
2.
36  81
OK, so what is a complex number?
A complex number has two parts – a real
part and an imaginary part.
A complex number comes in the form
real
a + bi
imaginary
And just so you know…
All real numbers are complex  3 = 3 + 0i
All imaginary numbers are complex 
7i = 0 + 7i
Again, treat the i as a variable and you will
have no problems.
Lets try these 4 problems.
1. (8  3i)  (6  2i)
2. (8  3i)  (6  2i)
3. (8  3i)  (6  2i)
4. (8  3i)  (6  2i)
More Practice
5. 6i
5
6 - i 4  2i
6.

4
3i