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Complex Numbers Class 11 Ch-04

Complex Numbers Chapter-4 Class-11

Basic points and complete explanation of the complex numbers with examples and diagrams. Addition, subtraction and multiplication of complex numbers. Representation of a complex number into polar form

Following concepts will be  discussed in this Blog

  • Real Numbers
  • Non-Real Numbers or Imaginary Numbers and concept of iota.
  • Complex Number and its general form.
  • Conjugate of a complex Number.
  • Properties (Addition, Subtraction and multiplication) of Complex Numbers.
  • Modulus and Argument of a Complex Number.
  • Polar form of a complex number.
  • Nature of Quadratic Equations and their solutions.
  • Square Root of a Complex Number with example.

Real numbers:
All rational and irrational numbers are called real numbers.
Non-Real Numbers : 
The numbers which are not real are called non real numbers. These numbers are also called imaginary numbers.
Imaginary numbers :-
Square root of a negative real number is called the imaginary number.

Complex Number:
All real and non- real numbers are called complex number
A number of the form a + ib is called a complex number. Here a and b are the real numbers and i is called the iota.

For example  Z = a + ib,  where a is called real part and b is called imaginary part. 
 Value of iota is √-1
i = √-1 ,  i2 = -1,   i3 = - i,   i4 = 1
Geometrical Representation of the values of iota.

Two complex numbers are equal if their real and imaginary parts are equal
If Y and Z are two complex numbers such that
Y = a + ib and Z = c + id,
Then  Y = Z 
    a = c and  b = d

Addition of two complex numbers 
Two complex numbers are added by simply adding their real and imaginary parts
If Y = a + ib and Z = c + id then
Y + Z = (a + c) + i(b + d)
Properties of addition of two complex numbers
i) Addition of two complex numbers holds the closure property. This means that addition of two complex numbers is also a complex number.

ii) Addition of complex numbers is commutative.
Y + Z = Z + Y

iii) Addition of complex numbers holds associative law
(X + Y) + Z = X + (Y + Z)

iv)Existence of additive identity: 
0 is the additive identity for the addition of complex number.  Z + 0 = Z

v) Existence of additive inverse : 
If Z is a complex number then - Z is called the additive inverse of Z.      Z + ( - Z) = 0
Subtraction of two complex numbers : 
Two complex numbers are subtracted by subtracting their corresponding real and imaginary parts. 
If Y = a + ib,   Z = c + id then 
                Y - Z = (a + ib) - (c + id)
                          = (a - c) + i(b - d)
Multiplication of two complex numbers 
If Y = a + ib,  Z = c + id then  
YZ = (a + ib)(c + id)
      = ac + iad + ibc + i2bd 
      = ac + i(ad + bc) + (-bd)
     = (ac - bd) + i(ad + bc)
Properties of Multiplication of two complex numbers
i) Closure property : 
Product of two complex numbers is also a complex number.

ii) Commutative law :
 Multiplication of complex numbers is commutative.  YZ = ZY

iii) Associative Law:
 Multiplication of complex numbers is associative.    (XY)Z = X(YZ)

iv) Existence of multiplicative identity: 
1 is called the multiplicative identity for the product of complex number.
Z x 1 = 1 x Z = Z

v) Multiplicative inverse :
 If Z is a complex number then Z-1 is called the inverse of Z.    Z x  Z-1  = 1 

vi) Distributive law :
 Product of complex numbers holds the distributive property.  X(Y + Z) = XY + XZ

Conjugate of a complex number \[If\: Z = a+ib \; is\; a\; complex\; number\; then \; conjugate\; of\; Z\; is \; denoted\; by\; \overline{Z}\; and \; is\; given\; by\]\[\overline{Z}=\overline{a+ib}=a-ib\]
Note:- Conjugate of the conjugate of complex number is the complex number itself. \[If\: z=a+ib,\: then\: \overline{z}=\overline{a+ib}=a-ib\]\[\overline{\overline{z}}=\overline{a-ib}=a+ib=z\: \Rightarrow \overline{\overline{z}}=z\]

Modulus of complex number
If Z = a + ib is a complex number then modulus of Z is denoted by |Z| and is given by \[\left | Z \right |= \sqrt{a^{2}+b^{2}}\]

Argand Plane : 
The plane having a complex number assigned to each of its point is called a complex plan or Argand plane.

Every point of the form (x,y) can be represented in the cartesian coordinate plane.

When a complex number is represented in the plan then the plane is called a complex plane or Argand plane.

In Argand plane x- axis is called real axis and y-axis is called imaginary axis.
If Z = x + iy then \[\left | Z \right |= \sqrt{x^{2}+y^{2}}\]
|Z| is the distance of the point (x,y) from the origin (0,0)

Polar Representation of a complex number z = x + iy

Let point P represent the complex number Z = x + iy. Let directed line OP = |Z| = r and θ is the angle which OP makes with the positive direction of x - axis. \[In\; \Delta OAP,\; \; \; \frac{OA}{OP}=cos\theta\]\[\frac{x}{r}=cos\theta \Rightarrow x=rcos\theta\]
\[In\; \Delta OAP,\; \; \; \frac{AP}{OP}=sin\theta\]\[\frac{y}{r}=sin\theta \Rightarrow y=rsin\theta\]
Putting x = rcosθ and y = rsinθ in   Z = x + iy we get
Z = rcosθ + irsinθ  or   Z = r(cosθ + isinθ)
This is called polar form of a complex number.

If Z = a + ib, then to convert it into polar form  we put a = rcosθ and b = rsinθ
tanθ = b/a  and     θ = tan-1(b/a)  

 Sign convention while converting a complex number from Cartesian form to polar form.



If (a, b) lie in first quadrant then Argument =   θ

If (a, b) lie in second quadrant then Argument = π - θ

If (a, b) lie in third quadrant then Argument =  -π + θ

If (a, b) lie in fourth quadrant then Argument =  - θ

If complex number Z is purely +ve real number then Argument is  0

If complex number Z is purely -ve real number then Argument is  π

If complex number Z is purely +ve imaginary number then Argument is  π/2

If complex number Z is purely -ve imaginary number then Argument is  - π/2

Method to convert a complex number into a polar form
\[z=-\sqrt{3}+i\]\[r=|z|=\sqrt{\left ( \sqrt{3} \right )^{2}+\left ( 1 \right )^{2}}=2\]Let polar form of complex number is \[z=rcos\theta +irsin\theta\]Putting r = 2 we get\[z=2cos\theta +i2sin\theta\]Compairing this with the first equation we get \[2cos\theta =-\sqrt{3}\; \; and\; \; 2sin\theta =1\]\[\Rightarrow cos\theta =-\frac{\sqrt{3}}{2}\; \; and\; \; sin\theta =\frac{1}{2}\]\[tan\theta =\frac{sin\theta }{cos\theta }=\frac{1/2}{-\sqrt{3}/2}=\frac{-1}{\sqrt{3}}\]\[tan\theta =\frac{-1}{\sqrt{3}}=-tan\left ( \frac{\pi }{6} \right )\]\[since\; point \; \; (-\sqrt{3},1)\; \; lie \; in \; the\; second\; quadrant\]\[\Rightarrow Argument = \pi -\theta =\pi -\frac{\pi }{6}=\frac{5\pi }{6}\] Hence required polar form of a complex number is \[z=2cos(\frac{5\pi }{6})+i\: 2sin(\frac{5\pi }{6})\]
Quadratics Equations
General quadratic equation is  ax+ bx + c = 0
Discriminant (D)  = b2 - 4ac

Nature Of The Roots Of The Quadratic Equation
\[If D> 0 \; then \; roots\; are \; real\; and\; unequal\; or\; distinct\; or\; different)\]
\[If D= 0 \; then \; roots\; are \; real\; and\; equal\]
\[If D< 0 \; then \; roots\; are \; not\; real\]
\[If D\geq 0 \; then \; roots\; are \; \; real\]
Quadratic Formula For Solving The Quadratic Equations
\[x=\frac{-b\pm \sqrt{D}}{2a}\; \; \; or\; \; \; x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\]

Method of finding the square root of a complex number

This can be explained by taking an example

\[Find\; the\; square \; root\; of\; the\; complex\; number\; \; z=\sqrt{3+i4}\]\[Let\; \; a+ib=\sqrt{3+i4}\]Squaring on both side we get,\[(a)^{2}+(ib)^{2}+2iab=3+i4\]\[(a)^{2}-b^{2}+2abi=3+i4\]Compairing real and imaginary parts we ger, \[(a)^{2}-b^{2}=3 .......(1),\: \: and\: 2ab=4\]\[Now\: using\: the\: formula\: \: (a^{2}+b^{2})^{2}=(a^{2}-b^{2})^{2}+(2ab)^{2}\]\[(a^{2}+b^{2})^{2}=(3)^{2}+(4)^{2}=9+16=25\]\[\Rightarrow (a^{2}+b^{2})=5 ......(2)\]Adding eqn(1) and eqn(2) we get\[2a^{2}=8\Rightarrow a^{2}=4\Rightarrow a=\pm 2\]From eqn(1) - eqn(2) we get\[2b^{2}=2\Rightarrow b^{2}=1\Rightarrow b=\pm 1\]\[\sqrt{3+i4}=a+ib=\pm 2\pm 1i=\pm (2+i)\]Here between 2 and i we apply +ve sign because 2ab is positive. If the value of 2ab is -ve then between a and b we apply -ve sign in the square root of the complex number.


Question:\[If\: x=-5+2\sqrt{-4}, \: then\: find \: the\: value\: of\\ x^{4}+9\: x^{3}+35\: x^{2}-x+4\]
Solution:\[x= -5+2\sqrt{-4}\]\[x+5=2\sqrt{-4}\]
Squaring on both sides we get\[(x+5)^{2}=\left [ 2\sqrt{-4} \; \right ]^{2}\]\[x^{2}+5^{2}+2\times x\times 5=4\times (-4)\]\[x^{2}+25+10x+16=0\]\[x^{2}+10x+41=0\]
Now divide the given equation with this equation we get

Dividend = Divisor X Quotient + Remainder
        P(x) = (x2 + 10x + 41)(x2 – x + 4) – 160
                = 0 X (x2 – x + 4) – 160 = - 160 Ans


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