Basics & Formulas Ch-4 Class11 | Complex Numbers

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

OR

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  and is given by 
 

Note:- Conjugate of the conjugate of complex number is the complex number itself.

If Z = a + ib, then 

 

Modulus of complex number

If Z = a + ib is a complex number then modulus of Z is denoted by |Z| and is given by   

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 

|Z| is the distance of the point (x, y) from the origin (0, 0)

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. 

 

 

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.

DISCUSSION AND CONCLUSIONS 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

Let polar form of complex number is 

 

Putting r = 2 we get

 

Compairing this with the first equation we get 

 

 

 

Since point lie i

 

Hence required polar form of a complex number is 

Quadratics Equations

General quadratic equation is  ax² + 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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