1. Complex Numbers

2. 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).

3. REAL NUMBERS
&
IMAGINARY NUMBERS

4. Real Numbers are numbers like:
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

5. 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.

6. The "unit" imaginary number is i, which is the square root of −1
Examples of Imaginary Numbers: 3i
1.04i
−2.8i
3i/4
(√2)i
1998i 1
12.38
−0.8625
3/4 √2
1998

7. STANDARD FORM OF A COMPLEX NUMBER-
A Complex Number is a combination of a  Real Number and an Imaginary Number

8. SQUARE ROOT OF NEGATIVE INTEGERS -

9. Examples of Comple Numbers
Examples of complex numbers:
Real Part
Imaginary Part a bi + 2 7i + 20 3i –
Real Numbers: a + 0i
Imaginary Numbers: 0 + bi
Simplify:
= i
= 3i 1.
= i
= 8i 2. + 3.
+ i =
a + bi form
+ i =
Simplify using the product property of radicals.
4 + 5i =
Examples of Comple Numbers

10. Add or Subtract Complex Numbers
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
Add or Subtract Complex Numbers

11. Adding Complex Numbers
Examples: Add (11 + 5i) + (8 – 2i )
= (11 + 8) + (5i – 2i )
Group real and imaginary terms.
= i
a + bi form
Add ( ) + (21 – )
= (10 + i ) + (21 – i )
i =
= ( ) + (i – i )
Group real and imaginary terms.
= 31
a + bi form
Adding Complex Numbers

12. Subtracting Complex Numbers
Examples: Subtract: (– i ) – (7 – 9i)
= (– 21 – 7) + [(3 – (– 9)]i
Group real and imaginary terms.
= (– 21 – 7) + (3i + 9i)
= – i
a + bi form
Subtract: ( ) – ( )
Group real and imaginary terms.
= (11 + i ) – (6 + i )
= (11 – 6) + [ – ]i
= (11 – 6) + [ 4 – 3]i
= 5 + i
a + bi form
Subtracting Complex Numbers

13. Product of Complex Numbers
The product of two complex numbers is defined as:
(a + bi)(c + di ) = (ac – bd ) + (ad + bc)i
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.
Product of Complex Numbers

14. Examples: 1. = i i = 5i i = 5i2 = 5 (–1) = –5

15. The complex numbers a + bi and a - bi are called conjugates.

16. 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
Product of Conjugates

17. 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.
Example:
Multiply the expression by . –1
Replace i2 by –1 and simplify.
Write the answer in the form a + bi.

18. Simplify:
Multiply the numerator and denominator by the conjugate of 2 + i.
In 2 + i, a = 2 and b = 1. a2 + b2 = –1
Replace i2 by –1 and simplify.
Write the answer in the form a + bi.
Example: (5 +3i)/(2+i)

19. 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

20. HOME ASSIGNMENT
* Express in the form of a + ib
(i) ( 5 – 3i ) ( 5 + 4i )
i ( 8 – 3i ) ( 5i )
(iii) 3( 7 + i7 ) + i (7 + i7 )
( iv) (1 – i) – ( –1 + i6 )
* Solve each of the following equations:
2x²+ x + 1
3x² +1 = 0