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Class – XI, CBSE MB1105: Complex Number and Quadratic Equations

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What is iota?

Class – XI MB1105: Complex Number and Quadratic Equations

Important Definitions

Topic: Complex Numbers



What is iota?

Class - XI MB1105: Complex Number and Quadratic Equations



2

We denote 1 by the symbol .

Therefore, we have 1.

We call the symbol as iota.

i

i

i



What are Complex Numbers?






What are Complex Numbers?




Any number of the form is called

a complex number, if and both are real

numbers.

(a) is called as the real part of ,

denoted by Re( )

(b) is called as the imaginary part of ,

denote

z a ib

a b

a z

z

b z

d by Im( ).

The set of complex numbers is denoted by .

If 0 and 0, the complex number

becomes 0 0 0, which is called the zero

complex number.

z

C

a b

i



What is Equality of Complex Numbers?






What is Equality of Complex Numbers?




Two complex numbers and

, are said to be equal, if

and .

a ib

c id

a c b d



What is addition of Complex Numbers?



Topic: Operations on Complex Numbers



What is addition of Complex Numbers?




1

2

1 2

Two complex numbers are added by adding their

respective real and imaginary parts. Thus, if

and are two complex numbers, then

z , which is again a complex

number.

We can al

z a ib

z c id

z a c i b d

so visualize the addition of complex numbers

in the complex plane as follows:



What is difference of two complex numbers?






What is difference of two complex numbers?




1

2 1 2

1 2 1 2

Given any two complex numbers

and , the difference is

defined as follows:

.

z

z z z

z z z z



What is multiplication of complex numbers?






What is multiplication of complex numbers?




1 2 1 2 2 2 1 2 1 12 2 1 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 2 1

We can multiply two complex numbers using the distributive

property as follows:

.

Thus, .

On the complex plane, th

z z a ib a ib a a a ib a ib ib ib

a a ia b ib a b b

z z a a b b i a b a b

e product of two complex numbers is

tough to determine till we learn about the polar representation

of complex numbers. But, for now, here is a visual that can help

you understand this a bit.



What is division of two complex numbers?






What is division of two complex numbers?




1 2 2

1

2

1

To divide two complex numbers, we

multiply the dividend by the

multiplicative inverse of the divisor.

Therefore, given any two complex

numbers and , where 0,

the quotient is defined as

z z z

z

z

z

z

12 2

1.z

z



What is conjugate of a complex number?



Topic: Conjugate of Complex Numbers



What is conjugate of a complex number?




The conjugate of a complex number

is defined by . On the complex planes

these numbers are the reflection of each other

with respect to the real (horizontal) axis.

z x iy

z x iy



What is modulus of a complex number?



Topic: Modulus of Complex Numbers



What is modulus of a complex number?




2 2

Modulus of a complex number

is denoted by mod or

and is defined as ,

where Re and Im .

We call as the absolute value

of . We also note that 0.

z a ib z z

z a b

a z b z

z

z z



What is Complex Plane?



Topic: Complex Plane







A complex plane or Argand plane provides a

visual representation of complex numbers. It

is a plane having a complex number assigned

to each of its point. It consists of a horizontal

axis called the real axis and a vertical axis

called the imaginary axis. Thus, a complex

plane is a modified Cartesian plane, with the

real part of a complex number represented

along the -axis, and the imaginary part x along

the -axis. Thus, the complex number

corresponds to the point ( , ) in the complex

plane.

y z x iy

x y


If a complex number is purely real, then imaginary part is zero. Therefore, a

purely real number is represented by a point on -axis. A purely imaginary

complex number is represented by a point on -a

x

y xis. That is why -axis is

known as the real axis and -axis as the imaginary axis.

x

y







In the Argand plane, the modulus of

the complex number is the distance

between the point to the origin as is

shown in the figure.

The points on the -axis corresponds to the complex numbers of the form

0 and the points on the -axis corresponds to the complex numbers

of the form 0 . The representation of a complex number

x

a i y

ib z

and

its conjugate in the Argand plane are, respectively, the points

, and , .

x iy

z x iy

P x y Q x y

Geometrically, the point , is

the mirror image of the point ,

on the real axis as is shown in the

figure.

x y

x y


What is polar representation of complex number?









Topic: Polar Representation of Complex Number


The polar coordinate system consists of

concentric circles centered at origin. Any

point ( , ) can be specified as a set of

coordinates ( , ) where is the distance

of the point from the origin and

a b

r r is the

angle that the radius to the point makes

with the positive direction of the

horizontal axis. (See image)

Any complex number can be

represented in its polar form by writing

cos and sin . The following

image explains how we get the above

relations.

x iy

x r y







2 2

We have cos sin . The latter is said to be

the polar form of the complex number . Here

is the modulus of , cos ,sin ,

and is called the argument (or amplitude) of and i

z x iy r i

z

x yr x y z z

r rz

s

denoted by arg .

For any non zero complex number there is only one value

of in 0 2 . However, any other interval of length

2 can also be taken as such an interval, for example

. The uniq

z

z

ue value of such that

for which cos and sin , is known as the

principle value of the argument of . The general value

of the argument is 2 , is an integer and is the

principle v

x r y r

z

n n

alue of arg . z







The following figures shows some of the possible arguments of a

complex number such that 0 2 and .

For 0 2 , we have the following:

z

For , we have the following:


What is Quadratic Equation?



Topic: Quadratic Equations



What is Quadratic Equation?





2An equation of the form 0 where 0 and , , are real

numbers, is called a quadratic equation. The number , , are called the

coefficients of the quadratic equation.

A root of the above quad

ax bx c a a b c

a b c

2

2

ratic equation is a number (real or complex)

such that 0.

The roots of the above quadratic equation are given by

2

where the quantity 4 is known as the discriminant of the

equat

a b c

b Dx

a

D D b ac

1 2

ion.

If 0, then quadratic equation has non real but complex roots, given by

and . ( Since 0, thus, and 2 2

he

D

b i D b i Dx x D

a a

1 2

nce ).

Clearly, , are complex conjugate of each other.

D i D

x x


Fact # 1

Every real number is a complex number so the system of complex numbers includes the system of real numbers.


Important Facts




Fact # 2

0 is both purely real and purely imaginary number.


Important Facts




Fact # 3

A complex number is an imaginary number if and only if its imaginary part is non zero. Here real part may or may not be zero 4 + 3i is an imaginary number, but not purely imaginary.


Important Facts




Fact # 4

All purely imaginary number except zero are imaginary numbers, but an imaginary number may or may not be purely imaginary.


Important Facts




Fact # 5

Complex numbers have 2 real dimensions, whereas the real numbers have only one dimension. Thus, while dealing with real numbers, we usually move on the real line, whereas while dealing with complex numbers, we move on a plane (a 2-D object).


Important Facts




Fact # 6


Important Facts



1 2

The addition of complex numbers satisfy the following properties:

(a) The sum of two complex numbers is a complex number i.e., is a complex number

z zThe closure law :

1 2

1 2 1 2 2 1

1 2 3 1 2 3 1 2 3

for all complex numbers and .

(b) For any two complex numbers and , .

(c) For any three complex numbers , , , .

z z

z z z z z z

z z z z z z z z z

The commutative law :

The associative law :

(d) There exists the complex numbers 0 0 denoted as 0 , called the

additive identity or the zero

iThe existence of additive identity :

complex number, such that, for every

complex numbers , 0 .

(e) For every complex number ,

z z z

z a ib

The existence of additive inverse :

we have the complex number

(denoted as ), called the additive inverse or negative

a i b z

of , such that 0.z z z


Fact # 7


Important Facts



1 2(a) The product of two complex numbers is a complex number, the product is a complex number

z z

The multiplication of complex numbers satisfy the following properties :

The closure law :

1 2

1 2 1 2 2 1

1 2 3 1

for all complex numbers and .

(b) For any two complex numbers and ,

(c) For any three complex numbers , , ,

z z

z z z z z z

z z z z z

The commutative law :

The associative law : 2 3 1 2 3 .

(d) There exists the complex number 1 0 (denoted as 1), called the

m

z z z z

i

The existence of multiplicative identity :

ultiplicative identity such that .1 , for every complex number .

(e) For every non-zero complex number , we have the complex number

z z z

z a ib

The existence of multiplicative inverse :

12 2 2 2

1 denoted by or called the multiplicative inverse of

a bz z

a b a b z

1 2 3

1 2 3 1 2 1 3

1 2 3 1 3 2 3

1 such that . 1 (here, 1 is the multiplicative identity).

(f) For any three complex numbers , , ,

(a)

(b) .

zzz z z

z z z z z z z

z z z z z z z

The distributive law :


Fact # 8

While reducing a complex number to polar form, we always take the principle value of the argument.


Important Facts




Fact # 9

The fundamental theorem of algebra states that every non-constant single-variable polynomials with complex coefficients has at least one complex root.


Important Facts




Fact # 10

Every polynomial equation of degree n has n roots.


Important Facts




Fact # 11


Important Facts



2If in the quadratic equation , , , and is one root of the

quadratic, then the other root must be the conjugate and vice-versa

, , 0 .

ax bx c a b c R p iq

p iq

p q R q



Important Formulae


1. iIntegral powers of :

2

3 2 4 2 2

2 2

2 1 2

We have 1, 1. Therefore,

1 , 1 1 1.

We note that for any

1, when is even(a) 1 .

1, when is odd

, when is even(b) 1

, when i

n nn

nn n

i i

i i i i i i i i

n N

ni i

n

i ni i i i

i n

. s odd

1Also, for any , the value of is found out by writing this as and solving .

Thus, any integral power of can be expressed in terms of 1 or .

n nn

n N i ii

i i




Important Formulae


2. Identities for complex numbers :

1 2

2 2 21 2 1 2 1 2

2 2 21 2 1 1 2 2

3 3 2 2 31 2 1 1 2 1 2 2

3 3 2 2 31 2 1 1 2 1 2 2

2 21 2 1 2 1 2

For all complex numbers and we get the following:

(a) 2 .

(b) 2 .

(c) 3 3 .

(d) 3 3 .

(e) .

z z

z z z z z z

z z z z z z

z z z z z z z z

z z z z z z z z

z z z z z z




Important Formulae


3. Properties of conjugate :

2 2

(a) ; (b) 2Re ;

(c) 2 Im ; (d) Re Im ;

(e) if and only of is purely real;

z z z z z

z z i z zz z z

z z z

1 2 1 2 1 2 1 2

1 12

2 2

(f) if and only if is purely imaginary;

(g) ; (h) ;

(i) , where 0.

z z z

z z z z z z z z

z zz

z z




Important Formulae


4. Properties of modulus of a complex number :

1 2

1 2 1 2

1 12

2 2

For any two complex number and , we have

(a)

(b) provided 0.

z z

z z z z

z zz

z z



Method to find the multiplicative inverse of a non zero complex number x + iy:

Class – XI,CBSE

Important Procedures



MB1105: Complex Number and Quadratic Equations


Method to find the multiplicative inverse of a non zero complex number x + iy:

Class – XI,CBSE




2 2 2 2 2

2 2 2 2

1 1Multiplicative inverse of

.

x iy x iy x iyx iy

x iy x iy x iy x i y x y

x yi

x y x



Method to write a complex number in the form A + iB:

Class – XI,CBSE





a ib

c id


Method to write a complex number in the form A + iB:

Class – XI,CBSE




2 2 2 2 2 2

2 2 2 2

We have

, where and .

a ib c ida ib

c id c id c id

ac bd i bc ad ac bd bc adi

c d c d c dac bd bc ad

A iB A Bc d c d


a ib

c id


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