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CBSE Assignments class 09 Mathematics

  Mathematics Assignments & Worksheets  For  Class IX Chapter-wise mathematics assignment for class 09. Important and useful extra questions strictly according to the CBSE syllabus and pattern with answer key CBSE Mathematics is a very good platform for the students and is contain the assignments for the students from 9 th  to 12 th  standard.  Here students can find very useful content which is very helpful to handle final examinations effectively.  For better understanding of the topic students should revise NCERT book with all examples and then start solving the chapter-wise assignments.  These assignments cover all the topics and are strictly according to the CBSE syllabus.  With the help of these assignments students can easily achieve the examination level and  can reach at the maximum height. Class 09 Mathematics    Assignment Case Study Based Questions Class IX 

Complex Numbers Chapter 5 Class 11

Complex Numbers Chapter-5 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
https://dinesh51.blogspot.com

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 \[If\: Z = a+ib \; is\; a\; complex\; number\; then \; conjugate\; of\; Z\; is \; denoted\; by\; \overline{Z}\; and \; is\; given\; by\]\[\overline{Z}=\overline{a+ib}=a-ib\]
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.

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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
\[z=-\sqrt{3}+i\]\[r=|z|=\sqrt{\left ( \sqrt{3} \right )^{2}+\left ( 1 \right )^{2}}=2\]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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