Download Introduction to Complex Numbers

Introduction to Complex Numbers
Perquisites: Algebra
Look at the graph of the square root function
x
x
What is interesting about this graph? Well, notice that for values of x less than zero,
there is no red line. In fact, in the real world, the square root function is not defined for
numbers less than zero. However, mathematicians defined the square root function for
negative numbers anyway; they call these numbers imaginary numbers. The definition of
an imaginary number is
1  i , or  1  i 2
where i is the imaginary constant. The imaginary constant behaves just like a real
constant. For example, the square root of -25 is
 25   1 25   1  25  i  25  5i
Follow this process, and find the values of the square roots of the following.
1.
2.
3.
4.
-1
-225
-16
-144
Now, a complex number is simply a real number added to an imaginary number. You
can write a complex number as
a+bi
Where a and b are real numbers. Creating complex numbers actually allows us to create
special solutions to quadratic equations. By now, you should know that the equation
 b  b 2  4ac
. If b 2  4ac is less than 0,
2a
you can see that the radical becomes an imaginary number. For example
ax 2  bx  c  0 has solutions in the form
3x 2  4 x  2  0 has the solutions
 4  4 2  (4  3  2)  4   8
2 i 2 2 i 2

 
, 
23
6
3
3
3
3
Using this method, find the solutions of the following
5.
6.
7.
8.
9.
2x 2  2x  2  0
x2 1  0
7x2  7x  3  0
x 2  4x  5  0
x 2  3x  5  0
Arithmetic with complex numbers is similar to arithmetic with real numbers. For
example, to add two complex numbers, say 3+i and 7-2i, just add the terms together
(3+i) + (7-2i) =
(3+7) + (i-2i) =
10-i
You are just treating i like a variable! Do a similar thing for subtraction
(3+i) - (7-2i) =
(i+2i) + (3-7) =
3i-4
Multiplication is a little tricky; you need to use the distributive property
(3  i )  (7  2i ) 
(3  7)  (3  2i )  (i  7)  (i  2i ) 
21  6i  7i  2i 2 
21  i  2i 2
Remembering that  1  i 2 , you can find that
21  i  2 
19  i
Try solving the following problems
10.
11.
12.
13.
14.
15.
16.
17.
(3  2i )  (6  i )
(7  4i )  (5  2i )
(4  i )  (6  3i )
(9  5i )  (7  2i )
(8  i )  (10  2i )
(3  4i )  (7  2i )
(2  i)  (5  6i)
(5  9i)  (6  8i )
Multiplication might be a little tricky, but it does not come close to the next operation:
division. An important thing to know before we get into this is that paired off with every
complex number a+bi is another complex number, a-bi. In this situation, a-bi is called
the complex conjugate of a+bi. For example, the complex conjugate of 3+2i is 3-2i. You
just change the sign of the imaginary part of the complex number!
Now, the complex conjugate would not be special, except for one important thing. A
complex number multiplied by its complex conjugate will form a real number. To
see why this works, use the example a+bi
(a  bi)  (a  bi) 
a 2  abi  abi  b 2 i 2 
a 2  b 2i 2 
a2  b2
What happened to the imaginary constant? Hmm...
How can we take advantage of this fact in division? Well, let us look at a complex
fraction.
3
4i  2
This is meaningless, since we have no idea of the behavior of a complex number in the
denominator of a fraction. How can we get rid of the complex number in the
denominator, though? Try using the complex conjugate!
3
 4i  2


4i  2  4i  2
3(4i  2)

(4i  2)( 4i  2)
 12i  6

22  42 i 2
 12i  6

4  16
3 3
i
8 5
That was neat
Solve
18.
19.
10
1  3i
4
5i
7
20.
2i
5
21.
1  2i
22. (Challenge)
9  2i
3  4i
Now we go on to the complex plane! The complex plane is a common way to graph
complex numbers. Look
4+2i
2.5
Imaginary part
2
4
1.5
1
0.5
0
0
0
4
Real part
Do you see how this works? To graph a complex number, in this case 4+2i, you just do it
the way you would normally on graph paper; your y value is the imaginary part of the
number, and your x value is the real part. Graphing 4+2i is just like graphing the point
(4, 2)! Try graphing the following values on your own “complex plane”
23. 6-7i
24. 5+2i
25. 3i-6
17
26.
4i
27. 4i
Another interesting thing about the complex plane is that it redefines the absolute value
of a number. To find the absolute value of a complex number, just find the length of the
line from the origin to the point that represents that complex number. How can you find
this distance, though? Easy! Just use Pythagoreans theorem for the coordinates of the
complex number. For example, the complex number 3+4i has an absolute value of five,
since 3 2  4 2  5 2 . Try doing this for problems 23-27!
Concept questions
1. Find the square root of -169.
26
2. What does
equal?
3  2i
3. Find the sum, product, and difference of 5+7i and 6-2i.
4. What is the complex conjugate? Why is it special?
5. What are the complex solutions of 5 x 2  6 x  2  0 ?
Practice questions
Find the square root of the following
1.
2.
3.
4.
5.
6.
-121
-81
-196
-256
-14400
-16
What are the values of x?
7. 5 x 2  3x  1  0
8. 5 x 2  6 x  2  0
9. 10 x 2  10 x  3  0
10. 8 x 2  12 x  5  0
11. 6 x 2  5 x  3  0
12. 7 x 2  8 x  6  0
Find the product and sum
13. 4+2i and 3+7i
14. 21+4i and 9+8i
15. 8-2i and 3+5i
16. 6+i and 1-7i
17. 9+17i and 8+i
18. 4+2i and 3+7i
Find the quotient
19.
20.
21.
22.
23.
25
3  4i
34
5  3i
7
5  6i
13
2i  3
13  i
i5
Graph the following on a labeled coordinate grid
24. 4+ 3i
25. 7 + 2i
26. -6 + 5i
27. 2i
28. 6