Name: COMPLEX NUMBERS
Uploaded: 2021-12-15
Duration: 6 min 47 s
Description: Ppt Minimalistic Geometric Background, Ppt Background, Ppt ....

Ppt Minimalistic Geometric Background, Ppt Background, Ppt .... 

Scene 1

Ppt Minimalistic Geometric Background, Ppt Background, Ppt ...

Download Free png Maths teacher training | Shotton Hall SCITT ...

Maths Png & Free Maths.png Transparent Images #20180 - PNGio

Students will be able to identify the complex number . They will be able to apply the algebra of complex number They will remember the properties of conjugate and modulus of complex numbers They will be able to find out the square roots of complex numbers and solve the quadratic equations They will represent the complex numbers in polar form.. 

Scene 2

Students will be able to identify the complex number . They will be able to apply the algebra of complex number They will remember the properties of conjugate and modulus of complex numbers They will be able to find out the square roots of complex numbers and solve the quadratic equations They will represent the complex numbers in polar form.

KEY CONCEPTS. Integral powers of iota. Introduction of complex number. Algebra of complex number. Conjugate and its properties. Modulus and its properties. Geometrical Representation of a complex number. Argument or Amplitude of a complex number. Square root of a complex number.. 

Scene 3

Integral powers of iota. Introduction of complex number. Algebra of complex number. Conjugate  and its properties. Modulus  and its properties. Geometrical Representation of a complex number. Argument or Amplitude of a complex number. Square root  of  a complex number.

Previous Knowledge Testing Strategy. The following questions will be asked to test the previous knowledge of the students : , - Find a natural no. satisfying 2 x + 3 = 8 Find a rational no. satisfying 2 x + 3 = 8 Find x in R , if - 4 = 0 Find x in R , if + 4 = 0. 

Scene 4

The following questions will be asked to test the previous knowledge of the students :      ,  - Find a natural no. satisfying  2 x   +  3  = 8 Find a rational no. satisfying  2 x   + 3 = 8 Find  x  in R , if        -  4 = 0 Find  x  in R , if        + 4 = 0

Download Free png Maths teacher training | Shotton Hall SCITT .... 

Scene 5

2 × 2 =  4 (−2) × (−2) =  4 0 × 0 =  0 If we  multiply a number with itself result is always  positive, or zero .  It seems like we cannot multiply a number by itself to get a negative answer but  imagine  that there is such a number (call it  i  for imaginary) that could do this: 				      i     x     i   =  -1 Well, by taking the square root of both sides we get this :        = -1 Which means that   i    is the answer to the square root of  −1.

Scene 6

N.B  :  Euler  was the first to introduce the symbol   i Which is actually  very useful  because by simply  accepting  that  i   exists we can solve things that need the square root of a negative number.           N.B  :   		     ,  x  is a positive real no.

Scene 7

i   is the symbol with the property that                 is called imaginary unit. It is not a  real number. We can also show this by the method of  contradiction .  If 		  then we know that  i   > 0,  i  =  0   or  i  <  0 If we take 	                      (product of two positives.) That is, -1 > 0 which is false. Therefore,  i   cannot greater then 0. Similar contradiction can be arrived at by supposing that if ,  i   = 0  and  i   < 0. Thus, we can conclude that  i  is  neither    positive nor zero nor negative. By the  Trichotomy  property, it follows that is  i   not a real number.

INTEGRAL POWER OF ‘ i ’. For any integer , where, k is an integer.. 

Scene 8

For any integer , where, k is an integer.

Scene 9

If  a  and  b  are real numbers, the number a  +  i b is a complex number, and it is said to be written in standard form.  Let  a complex number z = a +  i b . Here, a = Real part of z = Re(z)  b = Imaginary part of z =  Img (z) If  b  = 0, the number a  +  i b  =  a is a  real number If   a = 0, the number a  +  i b   is called an imaginary number.

Scene 10

Real numbers and Imaginary numbers are subsets of the sets of complex numbers.

Scene 11

Write the complex number in standard form  							      a +   i b where a   = 1 and  b   =

Scene 12

Let  z 1   =  a +  i b   and  z 2  =  c +  i d     are two complex  numbers. Then, z 1  = z 2 ,   iff ,  Re(z 1 ) = Re( z 2 ) and  Img (z 1 ) =  Img (z 2 ) That is  iff ,  a = c and b = d

Scene 13

ADDITION AND SUBSTRACTION OF COMPLEX NUMBER

If    a +  i b and  c  +  i d   are two complex numbers written in standard form, their sum and difference are defined as follows : Sum  = (a +  i b) + (c +  i d) = ( a+c ) +  i ( b+d ) Difference  = (a +  i b) - (c +  i d) = (a-c) +  i (b-d) Ex: (3 + 2 i ) – (6 + 13 i )  	   = 3 + 2 i    –  6 – 13 i 	    = –3 – 11 i

Scene 14

(a +  i b)  (c+  i d) = (ac-  bd ) +  i (ad+  bc ) Note :  Multiplying complex numbers is similar to multiplying polynomials and combining like terms.    Ex:	 (6 – 2 i )(2 – 3 i ) ‏   		= 6 (2 – 3 i ) ‏  - 2 i   ( 2 – 3 i  ) ‏  		 = 12 – 18 i  – 4 i  + 6 i 2	    		=   12 – 22 i  + 6(-1) ‏  		= 6 – 22 i

Scene 15

Scene 16

Scene 17

Let z = x +   i y;  x,y   ϵ   R,  Conjugate of z is : Properties of Conjugate 		  ,  iff  z is purely real  	            ,  iff  z is purely imaginary

Scene 18

MULTIPLICATIVE INVERSE OF A COMPLEX NUMBER :

If z  ≠  0  , then multiplicative inverse of z is :

Scene 19

Let z = x +   i y;  x,y   ϵ   R,  Modulus of z is : Properties of Modulus

Scene 20

Amplitude / Argument of a  complex  number and its polar  form :-

Argand  plane  or  Complex plane   : The plane having a complex number assigned to each of its point is called  complex plane  or  Argand  plane .  The   x-axis  and  y-axis  in the  Argand  plane  are called  real  axis   and  imaginary axis   respectively. Let z = x +  i y  is a complex number.  The co-ordinates of  P( x,y ) and

Scene 21

8
COMPLEX PLANE
Imagioaryaxis.•e
Real axis
X-AXIS

Scene 22

Let z = x +   i y , then P( x,y ) = P(      )  where, P(      ) is the polar co-ordinate of the point) Let,    x = r  cos       , y = r sin    So, z= r( cos     +  i   sin   ) is the polar form of z. ‘    ’ is the  arg (z) or amp(z), where tan     =   Principal  argument  of  z is    ,  where

Scene 23

Position Of z   Principal   arg (z) =  Positive  x-axis 0 Positive  y-axis Negative x-axis Negative y-axis 1 st  Quadrant 2 nd  Quadrant 3 rd  Quadrant 4 th  Quadrant

Scene 24

Let z = x +   i y , the square root of  is  Let, Therefore,  x = a 2   - b 2  and y = 2ab On solving  a and b we can find square root of z.

Scene 25

Complex numbers has a wide range of application in  Science ,  Engineering ,  Statistics ,  etc . Applied mathematics  : Solving different equations with function of complex roots. Cauchy’s Integral Formula Calculus of residues In Electric Circuits  : To solve electric circuits

Scene 26

HOW COMPLEX NUMBERS CAN BE APPLIED TO “The real world”

Examples of the application of complex numbers: Electric Field and Magnetic Field. Application in Ohm’s Law. In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes. A complex number could be used to represent the position of an object in a two  dimensional plane.

CONCEPT MAPPING. Set of Imaginary No.. Set of Real No.. 

Scene 27

Complex No. a+ib a = Real Part b = Imaginary Part

Product of  z 1 z 2 =(ac- bd )+ i ( ad+bc )

Polar Form z =  a+ib = r( cos    + i  sin  ) where, r =              , modulus of z cos    =     sin   =

A polynomial equation with degree n, has n roots. ax 2  + bx+c =0 ; a ≠ 0, b-4ac<0 x=

COMPLEX NUMBERS

Page 1 (0s)

Page 2 (14s)

Page 3 (33s)

Page 4 (49s)

Page 5 (1m 8s)

Page 6 (1m 33s)

Page 7 (1m 50s)

Page 8 (2m 21s)

Page 9 (2m 32s)

Page 10 (2m 51s)

Page 11 (3m 2s)

Page 12 (3m 11s)

Page 13 (3m 24s)

Page 14 (3m 40s)

Page 15 (3m 56s)

Page 16 (4m 3s)

Page 17 (4m 9s)

Page 18 (4m 21s)

Page 19 (4m 30s)

Page 20 (4m 40s)

Page 21 (5m 0s)

Page 22 (5m 7s)

Page 23 (5m 24s)

Page 24 (5m 36s)

Page 25 (5m 49s)

Page 26 (6m 3s)

Page 27 (6m 23s)