Download presentation

Presentation is loading. Please wait.

Published byJeffery Randall Modified over 3 years ago

1
An Introduction

2
Any number of the form x + iy where x,y are Real and i=-1, i.e., i 2 = -1 is called a complex number. For example, 7 + i10, -5 -4i are examples of complex numbers. The set or complex numbers is denoted by C, i.e., C={x+iy: x,y belongs to R(set of Real number) and i=-1}. The complex number x+yi is usually denoted by z; x is called real part of z and is written as Re(z) and y is called imaginary part or z and is written as Im(z). So, for every z belongs to C. Z=Re(z) + i Im(z). For example, if z= 3 +5i, then Re(z)=3 and Im (z)=5.

3
Addition of complex numbers Let z 1 =x 1 +iy 1 and z 2 =x 2 +iy 2 be any two complex numbers, then the sum or addition of z 1 and z 2 is defined as (x 1 +x 2 ) + i(y 1 +y 2 ), and it is denoted by z 1 +z 2. The sum or two or more complex numbers is also a complex number. Negative of a complex number Let z=x+iy be any complex number, then the number –x-iy is called negative of z and is denoted by –z. Negative of a complex number is again a complex number.

4
Difference of Complex numbers Let z 1 =x 1 +iy 1 and z 2 =x 2 +iy 2 be any two complex numbers, then the difference of z 2 from z 1 is defined as (x 1 -x 2 ) + i(y 1 -y 2 ), and it is denoted by z 1 -z 2. The difference of two complex numbers is also a complex number.

5
Multiplication of complex numbers. Let z 1 =x 1 +iy 1 and z 2 =x 2 +iy 2 be any two complex numbers, then the product or multiplication of z 1 and z 2 is defined as (x 1 x 2 - y 1 y 2 ) + i(x 1 y 2 + y 1 x 2 ) and it is denoted by z 1 z 2. For example: z 1 = 5+3i and z 2 = 2+4i then z 1 z 2 = (10-12) +i(20+6)=-2+26i.

6
Lets understand the entire process with the help of an example. Find the square roots of 3-4i Solution: let x+iy be a square root of 3-4i Then (x+iy) 2 =3-4i x 2 -y 2 +2xyi=3-4i ( (a+b) 2 =a 2 +b 2 +2ab) => x 2 –y 2 =3 ………………..(i) And 2xy=-4, i.e., xy=-2………..(ii) We have (x 2 +y 2 ) 2 = (x 2 -y 2 ) 2 +4x 2 y 2 3 2 +4(4)=9+16=25 x 2 +y 2 =+5 but as x 2 +y 2 =5………………..(iii) On adding (i) and (iii), we get 2x 2 =8 => x 2 =4 => x=+2 Substituting in (ii), we get x=2, y=-1 and x=-2, y=1. Therefore, the two square roots of 3-4i are 2-i and -2+i.

7
Let x be a cube root of unity, then x 3 =1. => x 3 -1=0=> (x-1)(x 2 +x+1)=0 => either x-1=0 or x 2 +x+1=0 Either x=1or x=(-1+-3)/2 Thus, the three cube roots of unity are 1, (-1+i3)/2 and (-1-i3)/2.

8
Either of the two non-real cube roots of unity is the square of the other. If one of the non-real cube root of unity is denoted by w(read it as omega), then the other is w 2. Further, to avoid any possible confusion, we shall take w= (-1+i3)/2 Thus, the three cube roots of unity are 1,w and w 2. Sum of the three cube roots of unity is zero. Thus 1+w+w 2 =0 Product of the cube roots of unity is one. i.e. 1.w.w 2 =w 3 =1 Either of the two non-real cube roots of unity is reciprocal of the other. Since w 3 =1, therefore, w.w 2 =1=> w and w 2 are reciprocals of each other.

9
When we write the complex number in polar form, then z=r(cos φ + i sin φ ) Then according to De-Moivres Theorem z n = [r(cos φ + i sin φ )] n = r n (cos n φ + i sin n φ )

10
If the Discriminant of any quadratic equation <0 i.e. –ve number. The roots of that quadratic equation are not real, that means the roots of such quadratic equation are complex number. If d<0where d=b 2 -4ac Then roots are (–b+b 2 -4ac)/2a.

11
Solve the equation 3x 2 +7=0 Here, the discriminant=0 2 -4x3x7=-84 Thereforex=(-0+-84)/2x3 =(+2(21)i)/6 => (+(21)i)/3

Similar presentations

OK

Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and b are real numbers): Example: Solve.

Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and b are real numbers): Example: Solve.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

File type ppt on cybercrime Ppt on human resources accounting Ppt on world diabetes day circle Free ppt on forest society and colonialism in india Ppt on automobile related topics on personality Download ppt on water conservation in india Ppt on democratic rights class 9 Ppt on world book day uk Ppt on periodic table of elements Ppt on life study of mathematician turing