Name: COMPLEX NUMBERS
Uploaded: 2021-12-15
Duration: 5 min 36 s
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Ppt Minimalistic Geometric Background, Ppt Background, Ppt .... 

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

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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.. 

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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.. 

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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. 

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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

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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.

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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.

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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.. 

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For any integer , where, k is an integer.

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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.

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

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

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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

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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

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(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

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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

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MULTIPLICATIVE INVERSE OF A COMPLEX NUMBER :

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

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

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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

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

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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

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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

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