Starts in the name of ALLAH

PRESENTATION BY

ANUM UROOJ Cs (104)Superior Group Of Colleges

Complex numbers

What is Complex

Numbers?

Modulus Conjugate

Polar Form

De Moivre's Theorm Subtraction

Multiplication

Addition

DivisionApplication

COMPLEX NUMBERS

Complex numbers

Real part Imaginary part

It can be written in the form :

Z = a + bi

A complex numbers is a number consisting a Real and Imaginary part.

COMPLEX NUMBERS

• SUBSETS

Real numbers and imaginary numbers are the subsets of the complex numbers

• Power of "i"

The power of i in complex numbers is equal to under root of negative one , can be written as

COMPLEX NUMBERS

COMPLEX NUMBERS

1. Real part "a" is drawn at x-axis

that is on vertically.

2. Imaginary part "b" is drawn at

y- axis that is on horizontally.

• Graphically Representation:

COMPLEX CONJUGATE

• The complex conjugate of complex number Z = a + bi, is denoted by

The complex number and its conjugate have the same real part.

Re(a) = Re ( )

The sign of the imaginary part of the conjugate complex is reversed.

Im(b) = Im -( )

Z = a - bi

COMPLEX CONJUGATE

• Graphicall Representaion

The conjugate is drawn at downward to Imaginary part that is downward to the x-

axis

Complex Modulus

• The modulus or magnitude of Z denoted by IZ , is the distance from the origin to the point (a,b).

Complex numbers

• Equal complex numbers

Two complex numbers are equal if their real parts are equal and their imaginary parts are equal.

If a + bi = c + di,Then,

a = c and b = c

ADDITION OF COMPLEX NUMBERS

• If a + bi and c + di are two complex numbers then addition of complex numbers are ,

(a + bi) + (c + di) = (a + c) + (b + d)i

• Example:

(2 + 4i) + (5 + 3i)

= (2 + 5) + (4 + 3)i

= 7 + 7i

Subtraction of complex numbers

• If a + bi and c + di are two complex numbers then subtraction of complex numbers are

(a + bi) - (c + di) = (a - c) + (b - d)i

• Example:

(3 + 2i) - (1+3i)= (3 - 1) + (2 - 3)i= 2 -1i= 2 - i

Multiplication of complex numbers • If a +bi and c + di are two

complex numbers multiplication of two complex numbers is

(a + bi)(c + di) = (ac -bd) + ( ad + bc)i

• Example:

(2 + 3i)(4 + 5i)

= (2x4 - 2x5) + (3x4 - 3x5)i

= (8 - 10) + (12 - 15)i

= -2 - 3i

DIVISION OF A COMPLEX NUMBERS

• If a + bi and c + di are two complex numbers then division of a complex numbers are

• Example:

EXPRESSING COMPLEX NUMBERS IN POLAR FORM

x = r cos 0 y = r sin 0

Z = r ( cos 0 + i sin 0 )

DE MOIVRE'S THEORM

DE MOIVRE'S THEORM is the theorm which show us how to take complex number to any power easily.

APPLICATIONS

• COMPLEX NUMBERS HAVE MANY APPLICATION IN SCIENCE, MATHEMATICS, ENGINEERING, STATICS ETC.

The complex equation is a basic formula used for designing air foils-airplane wings and Figuring out flow forces around a circular object in water for instance

A complex numbers could be used to represent the position of an object in a two dimentional plane

A complex number is used in solving diffrent equations with function of complex root

ANY QUESTION ??

Thankyou for your attention