Transcript 8.1 Complex Numbers

8.1
Complex Numbers
JMerrill, 2009
A Little History 
 Math is used to explain our universe. When a
recurring phenomenon is seen and can’t be
explained by our present mathematics, new
systems of mathematics are derived.
 In the real number system, we can’t take the
square root of negatives, therefore the
complex number system was created.
 Complex numbers revolutionized computer
graphics
Definition of the Imaginary Unit, i
i  (  1)  1
2
2
i  i  i  (1)i  i
3
2
i  (i )  (1)  1
4
2 2
2
Complex Numbers
 A complex number consists of a real and an
imaginary term:
Operations on Complex Numbers
 Add/subtract real to real, and imaginary to
imaginary
 Example:(6 + 7i) + (3 - 2i)

(6 + 3) + (7i - 2i) = 9 + 5i
 When subtracting, DON’T FORGET to
distribute the negative sign!
 Example: (3+2i) – (5 – i)

(3 – 5) + (2i – (-i)) = -2 + 3i
Operations on Complex Numbers
(2  3i)  (3  6i)
 6  12i  9i  18i2
 6  3i  18  ( 1)
 6  3i  18  24  3i
Dividing/Simplifying
 In order to simplify complex numbers (they
must always be in the form a + bi, you must
multiply by the complex conjugate:
3  8i (4  3i)
(3  8i)  (4  3i) 

4  3i (4  3i)
2
12  9i  32i  24i

16  9
12  41i 12 41i



25
25 25
You Do
 Simplify:
3  2i
4i
3  2i  3  2i   4  i  12  11i  2i2 10 11


 i


2
4 i  4 i  4 i 
17 17
16  i
Calculator 