### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

### 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

## Following concepts will be  discussed in this Blog

 Real NumbersNon-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

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)

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)

 DISCUSSION AND CONCLUSIONSIf (a, b) lie in first quadrant then Argument =   θIf (a, b) lie in second quadrant then Argument = π - θIf (a, b) lie in third quadrant then Argument =  -π + θIf (a, b) lie in fourth quadrant then Argument =  - θIf complex number Z is purely +ve real number then Argument is  0If complex number Z is purely -ve real number then Argument is  πIf complex number Z is purely +ve imaginary number then Argument is  π/2If 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 $z=2cos(\frac{5\pi }{6})+i\: 2sin(\frac{5\pi }{6})$
General quadratic equation is  ax+ bx + c = 0
Discriminant (D)  = b2 - 4ac

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