WELCOME TO OUR PRESENTATION

Presented By :Jakir Hasan Saniatul Haque

Presented To :Md. Arifuzzaman (AZ)Lecturer (Mathematics)Department Of Natural SciencesDaffodil International University

COMPLEX NUMBERS• An ordered pair of real number generally

written in the form “a+ib”

• Where a and b are real number and is an imaginary.

• In this expression, a is the real part and b is the imaginary part of complex number.

COMPLEX NUMBER

Real Number

Imaginary Number

Complex Number

Definition of pure imaginary numbers:

Any positive real number b,

where i is the imaginary unit and bi is called the pure imaginary number.

b2 b2 1 bi

Definition of pure imaginary numbers:

i 1

i2 1 i is not a variable

it is a symbol for a specific number

Simplify each expression.

1. 81 81 1 9i

2. 121x5 121x4 1 x 11x2i x

3. 200x 100 1 2x 10i 2x

Any number in form a+bi, where a and b are real numbers and i is imaginary unit.

Definition of Complex Numbers

COMPLEX NUMBER

Complex number extend the concept of one-dimensional number line to the two-dimensional complex plan.

• Horizontal axis use for real part. • Vertical axis for the imaginary part.

Equations like x2=-1 do not have a solution within the real numbers

12 x

1x

1i

12 i Real no:

Imaginary no:

Why complex numbers are introduced???

THE POWERS OF i: then,1- If i

12 i ii 3 14 i ii 5

16 i ii 7 18 i .etc

iii

)53()12()51()32(

i83

Example

Addition : Complex number added by adding real part in real and imaginary part in imaginary.

(a + b) + (c + d ) = (a + c) + (b + d)

Fundamental Operations with complex number

Subtraction: Similarly, subtraction is defined

(a + b) - (c + d ) = (a - c) + (b - d) .

ii

ii

21)53()12()51()32(

Example

Multiplication:

The multiplication of two complex number is define by the following formula:

(a + b).(c + d ) =(ac - bd) + (b c + ad) Square of the imaginary unit is -1.

²== -1

ii

ii

1313)310()152(

)51)(32(

Example

Division:Division can be defined as:

𝑎+𝑏𝑖  𝑐+𝑑𝑖  =¿ () + (

EXAMPLE i

i2176

i

iii

2121

2176

22

2

21147126

iii

415146

i

5520 i

55

520 i

i4

How complex numbers can be applied to “The Real World”???

Examples of the application of complex numbers: 1) Electric field and magnetic field.2) Application in ohms law.

3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes

4) A complex number could be used to represent the position of an object in a two dimensional plane.