5. Complex Number and Quadratic Equations | Introduction Mathematics class 11
5. Complex Number and Quadratic Equations | Introduction Mathematics class 11
Introduction
Introduction
⇒First we know about the real number;
π√Real Number includes;
(i) Whole numbers: like 0, 1, 2, 3, 4.................................. etc.
(ii) Rational numbers: like 4/5, 0/6, 0.333........, etc.
(iii) irrational numbers: like π, √3, √2 etc.
We have learnt quadratic equation in previous class. The nature of quadratic equations is
D > 0, {Real and unequal roots}
D = 0, {Real and equal roots}
D < 0, {No Real roots, i.e. Imaginary root}
Look the following example
x2 + 3x + 5 = 0
a = 1, b = 3, c= 5
D =
= 32– 4 ×1 × 5
= 9 – 20
= –11
D < 0, {so equation has no real but imaginary roots}
Now we have to find the roots
Here both the value of x is an imaginary number, which is made by the composition of (i), symbol “i” is called iota. Such number is called complex number.
1. Imaginary Number: A number whose square is negative is known as an imaginary number.
Ex: , , , etc.
2. Complex number: Any number which is of the form of x + iy, where x and y are real number and i = is called a complex number.
Ex : 3 + i5, 2 – i3, 5 + i2 and 4 +i3 etc.
It is denoted by z i.e. z = x +iy, in which Re(z) = x and Im(z) = y
A complex number has two parts;
(I) real part Re(z) {∈ R}
real part : 2, 3, 5, and 4 or may be any real number.
(II) imaginary part Im(z) {Real number with i(iota)}
imaginary part: i, i2, i3, i4, and i5 etc.
Every Real number is a complex number if x∈ R and y ∈ R; such as
z = 3 ⇒ 3 +i0, x = 3, y =0
z = –3⇒ – 3 +i0, x = –3, y =0
z = 7 ⇒ 7 +i0, x = 7, y =0
3. See the following complex numbers
z = 3, z = i3, z = 4, z = i7
z = 3 and z = 4 are purely Real
z = i3 and z = i7 are purely Imaginary
Complex Number
4. Power of i (iota)
∈
Algebra of Complex Numbers
7. Algebra of Complex Numbers
(A) Addition of Complex numbers
Let z1 = a + ib and z2 = c + id be any two complex numbers. Then, the sum z1 + z2 is defined as follows:
z1 + z2 = (a+ib)+(c+id) = (a + c) + i (b + d), which is again a complex number.
Ex: z1 =2 + i3 and z2 = 7 + i5
z1+ z2 = (2+7) +i(3+5)
= 9+i8
Alternative Method:
z1+ z2 = 2+i3 +7+i5
= 2+7+i3+i5
= 9+i(3+5)
= 9+i8
(B) Subtraction of Complex Numbers
Given any two complex numbers z1 and z2, the difference z1 – z2 is defined as follows:
z1 – z2 = (a+ib) – (c+id) = (a–c)+i(b–d)
For example: z1= 4+i3, z2 =3 + i7
z1 – z2 =(4–3)+i(3–7)= 1–i4
Note: {4 and 3are like term and i3 and i7 are another like term}
Alternative Method:
z1 – z2 =(4+i3) – (3+i7)
=4+i3 – 3 – i7
=4– 3+i3 – i7
=1+i(3 – 7)
=1-i4
(C) Multiplication of Complex numbers:
z1 × z2 = (a +ib) (c+id)
=a(c+id) +ib(c+id)
= ac + iad + ibc +( –bd)
= ac –bd +i(ad+bc)
For example: z1= 2+3i, z2=5 +2i
z1× z2 = (2+3i)( 5 +2i)
= (2×5 – 3×2)+ i(2×2+3×5)
= 10 – 6 + i(4+15)
= 4 + 19i
Alternative Method:
z1× z2 = (2+3i)( 5 +2i)
=2( 5 +2i) +3i( 5 +2i)
= 10 + 4i + 15i + 6i2
= 10 + 4i + 15i + 6(–1)
= 10 – 6 + 19i
= 4 + 19i
(D)Some Important identities:
- (z1 + z2)2 = z12 + 2 z1 z2 + z22
- (z1 – z2)2 = z12 – 2 z1 z2 + z22
- (z1 + z2)3 = z13 + 3 z12 z2 + 3 z1z22 + z23
- (z1 – z2)3 = z13 – 3 z12 z2 + 3 z1z22 – z23
- z12 – z22 = (z1 + z2)( z1 – z2)
The Modulus and Congugate of a Complex Number
The Modulus and Congugate of a Complex Number
Let z = a + ib be a complex number.
Then, the modulus of z, denoted by |z|, is defined to be the non-negative real number
, i.e
Conjugate: The conjugate is where you chaange the sign in the middle of two terms like
and the congugate of z, denoted as;
The conjugate is where you change the sign in the middle of two terms like this:
d,;d;a;'
Polar representation of a complex number
Polar representation of a complex number:
Argand plane: The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane.
Polar form: Polar form of a complex number is another way to represent a complex number on argand plane.
if z = x + iy is any complex number then
In polar representation a complex number z is represented by two parameters r and θ. Where parameter r is the modulus of a complex number and θ is the angle with the positive direction of x - axis.
using pythagoros theorem
Here Z of modulus = r and θ is called the argument (or amplitude) of z which is denoted by arg z.
Principle arguments of z : The value of θ such that – π < θ ≤ π, called principal argument of z and is denoted by arg z.
The point (x,y) represent a normal cartesian coordinate. But in polar form this point is reoresented by a special coordinate system which is called polar coordinate having (r, θ).
This coordinate (r, θ) represents the each location of a point of a complex number.
So, We have there is a relation between polar cordinates and cartesian coordinates.
We know;
z = Re(z) + Img(z)
Then x-axis represent Re(z) and y-axis represent Img(z).
We consider Origin (0, 0) as pole.
For any complex number z = x + iy is represented as r (cosθ + i sinθ) as any point on complex plane. This is called polar representation of a complex number. Where θ is angle between r and x-axis.
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Mathematics Chapter List
1. Sets
2. Relations and Functions
3. Trigonometric Functions
4. Principle of Mathematical Induction
5. Complex Number and Quadratic Equations
6. Linear Inequalities
7. Permutations and Combinations
8. Binomial Theorem
9. Sequences and Series
10. Straight Lines
11. Conic Sections
12. Introduction to Three Dimensional Geometry
13. Limits and Derivatives
14. Mathematical Reasoning
15. Statistics
16. Probability
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