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Complex
Numbers
Complicated ????
Complex does not mean complicated.
It means two types of numbers, real and imaginary,which
together form a complex, just like a building complex
(buildings joined together).
2
REAL NUMBERS
&
IMAGINARY NUMBERS
Real Numbers are numbers like:
Nearly any number you can think of is a Real
Number!
1
12.38
−0.8625
3/4
√2
1998
Imaginary Numbers when squared give
a negative result
when we square a positive number ,we get a positive
result, and
when we square a negative number, then also we get
a positive result
But Imagine, there is such a number, which when
squared, gives a negative result
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
The "unit" imaginary number is i, which is the square
root of −1
i2 = −1
Examples of Imaginary Numbers:
3i
1.04i
−2.8i
3i/4
(√2)i
1998i
1
12.38
−0.862
5
3/4
√2
1998
STANDARD FORM OF A
COMPLEX NUMBERA Complex Number is a combination of a
Real Number and an Imaginary Number
SQUARE ROOT OF NEGATIVE
INTEGERS -
Examples of complex numbers:
Real Part
a
2
20
Real Numbers: a + 0i
+
+
–
Imaginary Part
bi
7i
3i
Imaginary Numbers: 0 + bi
Simplify: 1.  90 = i 90 = i 9 • 10 = 3i 10
2.  64 = i 64 = 8i
3. 16 +  50
= 16 + i  50 a + bi form
= 16 + i 25 • 2 Simplify using the product property of
= 4 + 5i 2
radicals.
To add or subtract complex numbers:
1. Write each complex number in the form a + bi.
2. Add or subtract the real parts of the complex numbers.
3. Add or subtract the imaginary parts of the complex numbers.
(a + bi ) + (c + di ) = (a + c) + (b + d)i
(a + bi ) – (c + di ) = (a – c) + (b – d )i
Examples: Add (11 + 5i) + (8 – 2i )
= (11 + 8) + (5i – 2i )
Group real and imaginary terms.
= 19 + 3i
a + bi form
Add (10 +  5 ) + (21 –
 5)
= (10 + i 5 ) + (21 – i 5 ) i =
1
= (10 + 21) + (i 5 – i 5) Group real and imaginary terms.
= 31
a + bi form
Examples: Subtract: (– 21 + 3i ) – (7 – 9i)
= (– 21 – 7) + [(3 – (– 9)]i
= (– 21 – 7) + (3i + 9i)
= –28 + 12i
Group real and
imaginary terms.
a + bi form
Subtract: (11 +  16 ) – (6 +  9 )
= (11 + i 16 ) – (6 + i 9 ) Group real and
= (11 – 6) + [ 16 – 9 ]i
= (11 – 6) + [ 4 – 3]i
=5+i
imaginary terms.
a + bi form
The product of two complex numbers is
defined as:
(a + bi)(c + di ) = (ac – bd ) + (ad + bc)i
1. Use the FOIL ( First..Outer..Inner..Last )
method to find the product.
2. Replace i2 by – 1.
3. Write the answer in the form a + bi.
Examples: 1.  25 •  5 = i 25 • i 5
= 5i • i 5
= 5i2 5
= 5 (–1) 5
= –5 5
2. 7i (11– 5i) = 77i – 35i2
= 77i – 35 (– 1)
= 35 + 77i
3. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2
= 12 + 4i – 21i2
= 12 + 4i – 21(–1)
= 12 + 4i + 21
= 33 + 4i
The complex numbers a + bi and a - bi
are called conjugates.
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The product of conjugates is the real number a2 + b2.
(a + bi) (a – bi) = a2 – b2i2
= a2 – b2(– 1)
= a2 + b2
Example: (5 + 2i) (5 – 2i) = (52 – 4i2)
= 25 – 4 (–1)
= 29
Dividing Complex Numbers
A rational expression, containing one or more complex numbers,
is in simplest form when there are no imaginary numbers
remaining in the denominator.
7  9i
Example:
6i
7  9i • i
i

Multiply the expression by .
6i i
i
2
7i  9i

6i 2
7i  3(–1)
Replace i2 by –1 and simplify.

6(–1)
3  7i
1 7

   i Write the answer in the form a + bi.
6
2 6
Simplify: 5  3i
2 i
5  3i • 2  i

2  i 2 i
Multiply the numerator and
denominator by the conjugate of 2 + i.
10  5 i  6 i  3 i 2 In 2 + i, a = 2 and b = 1.

a2 + b2 = 22 + 12
2 2  12
10  i  3(–1)

4 1
13  i

5
13 1
  i
5 5
Replace i2 by –1 and simplify.
Write the answer in the form
a + bi.
The Mandelbrot Set
The beautiful Mandelbrot Set (pictured
here) is based on Complex Numbers.
It is a plot of what happens when we take
the simple equation z2+c (both complex
numbers) and feed the result back
into z time and time again.
The color shows how fast z2+c grows, and
black means it stays within a certain range.
Here is an image made by zooming into the
Mandelbrot set
HOME ASSIGNMENT
* Express in the form of a + ib
(i) ( 5 – 3i ) ( 5 + 4i )
(ii) i ( 8 – 3i ) ( 5i )
(iii) 3( 7 + i7 ) + i (7 + i7 )
( iv) (1 – i) – ( –1 + i6 )
* Solve each of the following equations:
(i) 2x²+ x + 1
(ii) 3x² +1 = 0