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2.  You can’t take the square root of a negative number. If you
use imaginary units, you can!
 The imaginary unit is ‘i ’.
 i =
 It is used to write the square root of a negative number.
1
Property of the square root of negative numbers
 If r is a positive real number, then
r ri
Examples:
3 3i 4 4i i2
b
aIndex Radicand

3. i
:then,1-If i
12
i ii3
14
i ii5
16
i ii7
18
i .etc
For i n … divide n by 4…
 If n is evenly divisible by 4 then i n =1
 If the remainder is 1, then i n = i
 If the remainder is 2, then i n = -1
 If the remainder is 3, then i n = -i

4.  A number consisting of two parts, one real and one imaginary
 For real numbers a and b the number a + bi is a complex
number.
 If b is 0, the complex number reduces to a which is a pure
real number.
 If a is 0, the complex number reduces to bi which is a pure
imaginary number.
 In other words all numbers, real and imaginary, are in the
set of complex numbers.
The combination of real and imaginary numbers make up the
complex number system
bia
Real part Imaginary part

5.  All numbers can be expressed as complex numbers.
 The complex conjugate of a complex number, z = x + jy,
denoted by z* , is given by
z* = x – jy.
 Two complex numbers
 a + bi and c + di are equal , if a = c and b = d
i033 ii 606
Properties of Complex Numbers
 The following properties of real numbers hold for complex
numbers.
 Associative Properties of Addition and Multiplication
 Commutative Properties of Addition and Multiplication
 Distributive property of Multiplication over Addition

6. Imaginary Axis
Real Axis
 We modify the familiar coordinate system by calling the horizontal
axis the real axis and the vertical axis the imaginary axis.
 Each complex number a + bi determines a unique position vector
with initial point (0, 0) and terminal point (a, b).

7. i34
.
i52
.i22
.
i34
.

8. Absolute Value of a Complex Number
 The distance the complex number is from the origin on the
complex plane.
 If you have a complex number
the absolute value can be found using:
)( bia
22
ba
Examples
1. i52
22
)5()2(
254
29
2. i6
22
)6()0(
360
36
6

9. Addition and Subtraction of Complex
Numbers
 Add or subtract the real parts, then add or subtract the
imaginary parts.
 For complex numbers a + bi and c + di ,
 Examples
(10 4i) - (5 - 2i)
= (10 - 5) + [4 (-2)]i
= 5 + 6i
(4 + 6i) + (3 + 7i)
= [4 + (3)] + [6 + 7]i
= 1 + i
idbcadicbia
idbcadicbia

10. Multiplication of Complex Numbers
 Treat the i’s like variables, then change any that are not to
the first power
 For complex numbers a + bi and c + di,
 The product of two complex numbers is found by multiplying as
if the numbers were binomials and using the fact that i2 = -1.
 Example:-
ibcadbdacdicbia
)3( ii
2
3 ii
)1(3i
i31
1.
Ex: )26)(32( ii
2
618412 iii
)1(62212 i
62212 i
i226

11. i
i
i
i
Ex
21
21
*
21
113
:
)21)(21(
)21)(113(
ii
ii
2
2
4221
221163
iii
iii
)1(41
)1(2253 i
41
2253 i
5
525 i
5
5
5
25 i
i5

12. The End
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