E –LESSON PLAN SUBJECT MATHEMATICS CLASS 11

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     E –LESSON PLAN   SUBJECT MATHEMATICS    CLASS 11
Board - DAV
CLASS –XI
SUBJECT- MATHEMATICS
CHAPTER 5  :- Complex Number And Quadratic Equations


TOPIC:   COMPLEX NUMBERS & QUADRATIC EQUATIONS

PRE- REQUISITE KNOWLEDGE:-
All definitions and important terms related to the Quadratic equations ,integers, natural numbers, irrational numbers, from rational to real numbers class 9& class10
TEACHING AIDS:- Green Board, Chalk,  Duster, Charts, smart  .
METHODOLOGY:-  lecture method
OBJECTIVE:
  •  Explain  the symbolism and integral powers of i.

·         Explain the conjugate and modulus of a complex numbers.
·        Explain the rules Multiplications and Multiplication inverse of a complex numbers.
·        Explain the polar form of a complex numbers .
·        Working rules of Argument of non zero number
·        Quadratic equation with complex numbers.
PROCEDURE :-

Start the session by asking the questions related to the natural numbers, integers and quadratic equation. Now introduce the topic complex numbers step by step as follows.
Topic
Explanation
Introduction of integral powers of i & complex numbers

Complex Number
A combination of a real and an imaginary number in the form a + bi

a and b are real numbers, and
i is the "unit imaginary number" √(−1)

The values a and b can be zero.

These are all complex numbers:
• 1 + i
• 2 − 6i
• −5.2i (an imaginary number is a complex number with a=0)
• 4 (a real number is a complex number with b=0)

Imaginary Number
A number that when squared gives a negative result.

When we square a Real Number (multiply it by itself) we always get a positive, or zero, result. For example 2×2=4, and (−2)×(−2)=4 as well.

So how can we square a number and get a negative result? Because we "imagine" that we can.

The "unit" imaginary numbers (the same as "1" for Real Numbers) is √(−1) (the square root of minus one), and its symbol is i, or j.

properties of Complex Numbers

  1. If x, y are real and x + iy = 0 then x = 0, y = 0.


Proof:

Since, x + iy = 0 = 0 + i0, hence by the definition of equality of two complex numbers it follows that, x = 0 and y = 0.
  1. If x, y, p, q are real and x + iy = p + iq then x = p and y = q.


Proof:

Since x + iy = p + iq

Hence x − p = -i(y − q) ⇒ (x − p)2 = i2 (y − q)2

⇒ (x − p)2 + (y − q)2 = 0 (Since i2 = -1)          (1)

Since x, y, p, q are real, (x − p)2 and (y − q)2are both non-negative. Hence the equation (1) is satisfied if each square is separately zero. Hence,

(x − p)2 = 0 or x = p and (y − q)2 = 0 or y = q. 
  1. Like real numbers, the set of complex numbers also satisfies the commutative, associative and distributive laws i.e., if z1, z2 and z3 be three complex numbers then,

    1. z1 + z2 = z2 + z1 (commutative law for addition) and z1. z2 = z2. z1 (commutative law for multiplication).

    1. (z1 + z2+ z3 = z1 + (z2 + z3) (associative law for addition) and (z1 z2) z3 = z1 (z2 z3) (associative law for multiplication).

    1. z1(z2 + z3) = z1 z2 + z1 z3 (distributive law).

  1. The sum and product of two conjugate complex quantities are both real.


Proof:

Let z = x + iy be a complex number where x, y are real.

Then, conjugate of z is  = x − iy.

Now, z +   = x + iy + x − iy = 2x, which is real.

And z.   = (x + iy)(x − iy) = x2 − i2y2 = x2 + y2 which is also real. 
  1. If the sum and product of two complex quantities are both real then the complex quantities are conjugate to each other.


Proof:

Let, z1 = a + ib and z2 = c + id be two complex quantities where a, b, c, d are real and b ≠ 0,

d ≠0.

By hypothesis, z1 + z2 = a + ib + c + id = (a + c) + i(b + d) is real.

Hence b + d = 0 or d = -b

And z1. z2 = (a + ib)( c + id) = (ac − bd) + i(ad + bc) is real.

Hence ad + bc = 0 or −ab + bc = 0 (Since d = -b)

Or b(c − a) = 0 or c = a (Since b ≠ 0)

Hence z2 = c + id = a + i(-b) = a − ib =  , which proves that z1 and z2 are conjugate to each other. 
  1. For two complex quantities z1 and z2, show that


|z1+ z2 | ≤ |z1 | + |z2
Polar Form of a Complex Number
The polar form of a complex number is another way to represent a complex number. The form z=a+bi=a+bi is called the rectangular coordinate form of a complex number.

The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of rr andθθ where rr is the length of the vector and θθ is the angle made with the real axis.
From Pythagorean Theorem :
r2=a2+b2r2=a2+b2
By using the basic trigonometric ratios :
cosθ=arcosθ=ar and sinθ=brsinθ=br .
Multiplying each side by rr :
rcosθ=a  and  rsinθ=brcosθ=a  and  rsinθ=b
The rectangular form of a complex number is given by
z=a+biz=a+bi .
Substitute the values of aa and bb .
z=a+bi    =rcosθ+(rsinθ)i    =r(cosθ+isinθ)z=a+bi    =rcosθ+(rsinθ)i    =r(cosθ+isinθ)
In the case of a complex number, rr represents the absolute value or modulus and the angle θθ is called the argument of the complex number.
This can be summarized as follows:
The polar form of a complex number z=a+biz=a+bi is z=r(cosθ+isinθ)z=r(cosθ+isinθ) , where r=|z|=a2+b2−−−−−−√r=|z|=a2+b2 , a=rcosθ  and  b=rsinθa=rcosθ  and  b=rsinθ, and θ=tan−1(ba)θ=tan−1(ba) for a>0a>0 and θ=tan−1(ba)+πθ=tan−1(ba)+π or θ=tan−1(ba)+180°θ=tan−1(ba)+180° for a<0a<0 
Quadratic Equations and Roots Containing "i "
A quadratic equation is of the form ax2 + bx + c = 0 where a, b and c are real number values with a not equal to zero.
Imaginary or complex roots will occur when the value under the radical portion of the quadratic formula is negative. Notice that the value under the radical portion is represented by "b2 - 4ac". So, if b2 - 4ac is a negative value, the quadratic equation is going to have complex conjugate roots (containing "i "s). 
b2 - 4ac is called the discriminant.


Introduction of imaginary numbers and real numbers in complex numbers.
Basic Formulas of  Conjugate and modulus of a complex numbers.
Properties of conjugate and modulus .
Multiplicative inverse.
Polar form &its representation with modulus & arguments.
Square root of complex numbers.
Quadratic equations with complex numbers
















EXPECTED OUTCOMES:-
After studying this lesson students will be able  to explain the term of polar form , their modulus and arguments. Students should know the powers of complex number. Students should be able to solve the quadratic equation of complex number.
STUDENTS DELIVERABLES:-
 Review questions given by the teacher. Presentation on the topic complex number of polar form. Solve worksheets problems with examples, solve assignment given by the teacher.                  
ASSESSMENT TECHNIQUES:-
Class test , oral test worksheets and assignments .


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