MB1105 QR

In this document you can review important
• Definitions
• Facts
• Formulae
• Procedures
Class – XI, CBSE
MB1105: Complex Number and
Quadratic Equations
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What is iota?
Class – XI
Quadratic Equations
Important Definitions
Topic: Complex Numbers

What is iota?
Class - XI
Quadratic Equations
2
W e denote 1 by the sym bol .
T herefore, w e have 1.
W e call the sym bol as iota.
i
i
i

What are Complex
Numbers?
Class - XI
Quadratic Equations

What are Complex
Numbers?
Class - XI
Quadratic Equations
Any num ber of the form is called
a com plex num ber, if and both are real
num bers.
(a) is called as the real part of ,
denoted by R e( )
(b) is called as the im aginary part of ,
denote
z a ib
a b
a z
z
b z
d by Im ( ).
T he set of com plex num bers is denoted by .
If 0 and 0, the com plex num ber
becom es 0 0 0, w hich is called the zero
com plex num ber.
z
C
a b
i

What is Equality of
Complex
Numbers?
Class - XI
Quadratic Equations

What is Equality of
Complex
Numbers?
Class - XI
Quadratic Equations
T w o com plex num bers and
, are said to be equal, if
and .
a ib
c id
a c b d

What is addition of
Complex
Numbers?
Class - XI
Quadratic Equations
Topic: Operations on Complex Numbers

What is addition of
Complex
Numbers?
Class - XI
Quadratic Equations
1
2
1 2
T w o com plex num bers are added by adding their
respective real and im aginary parts. T hu s, if
and are tw o com plex num bers, then
z , w hich is again a com plex
num ber.
W e can al
z a ib
z c id
z a c i b d
so visualize the addition of com plex num bers
in the com plex plane as follow s:

What is difference
of two complex
numbers?
Class - XI
Quadratic Equations

What is difference
of two complex
numbers?
Class - XI
Quadratic Equations
1
2 1 2
1 2 1 2
G iven any tw o com plex num bers
and , the difference is
defined as follow s:
.
z
z z z
z z z z

What is
multiplication of
complex numbers?
Class - XI
Quadratic Equations

What is
multiplication of
complex numbers?
Class - XI
Quadratic Equations
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
W e can m ultiply tw o com plex num bers usin g the distributive
property as follow s:
.
T hus, .
O n the com plex 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 tw o com plex num bers is
tough to determ ine till w e learn about the polar representation
of com plex num bers. But, for now , here is a visual that can help
you understand this a bit.

What is division of
two complex
numbers?
Class - XI
Quadratic Equations

What is division of
two complex
numbers?
Class - XI
Quadratic Equations
1 2 2
1
2
1
T o divide tw o com plex num bers, w e
m ultiply the dividend by the
m ultiplicative inverse of the divisor.
T herefore, given any tw o com plex
num bers and , w here 0,
the quotient is defined as
z z z
z
z
z
z
1
2 2
1
.z
z

What is conjugate
of a complex
number?
Class - XI
Quadratic Equations
Topic: Conjugate of Complex Numbers

What is conjugate
of a complex
number?
Class - XI
Quadratic Equations
T he conjugate of a com plex num ber
is defined by . O n the com plex planes
these num bers are the reflection of each other
w ith respect to the real (horizontal) ax is.
z x iy
z x iy

What is modulus of
a complex
number?
Class - XI
Quadratic Equations
Topic: Modulus of Complex Numbers

What is modulus of
a complex
number?
Class - XI
Quadratic Equations
2 2
M odulus of a com plex num ber
is denoted by m od or
and is defined as ,
w here R e and Im .
W e call as the absolute value
of . W e also note that 0.
z a ib z z
z a b
a z b z
z
z z

What is Complex
Plane?
Class - XI
Quadratic Equations
Topic: Complex Plane

What is Complex
Plane?
Class - XI
Quadratic Equations
A com plex plane or Argand plane provides a
visual representation of com plex num bers . It
is a plane having a com plex num ber assig ned
to each of its point. It consists of a h orizontal
axis called the real axis and a vertical axis
called the im aginary axis. T hus, a com plex
plane is a m odified C artesian plane, w ith the
real part of a com plex num ber represente d
along the -axis, and the im aginary partx along
the -axis. T hus, the com plex num ber
corresponds to the point ( , ) in the com plex
plane.
y z x iy
x y
If a com plex num ber is purely real, then im aginary part is zero. T herefore, a
purely real num ber is represented by a p oint on -axis. A purely im aginary
com plex num ber is represented by a point on -a
x
y xis. T hat is w hy -axis is
know n as the real axis and -axis as the im aginary axis.
x
y

What is Complex
Plane?
Class - XI
Quadratic Equations
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?
Class - XI
Quadratic Equations

What is polar
representation of
complex number?
Class - XI
Quadratic Equations
Topic: Polar Representation of Complex
Number
The polar coordinate system consists of
concentric circles centered at origin. A ny
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

What is polar
representation of
complex number?
Class - XI
Quadratic Equations
Number
2 2
W e have cos sin . T he latter is said to be
the polar form of the com plex num ber . H ere
is the m odulus of , cos , sin ,
and is called the argum ent (or am plitud e) of and i
z x iy r i
z
x y
r x y z z
r r
z s
denoted by arg .
For any non zero com plex num ber there is only one value
of in 0 2 . H ow ever, any other interval o f length
2 can also be taken as such an interval, for exam ple
. T he uniq
z
z
ue value of such that
for w hich cos and sin , is know n as the
principle value of the argum ent of . T he general value
of the argum ent is 2 , is an integer and is the
principle v
x r y r
z
n n
alue of arg .z

What is polar
representation of
complex number?
Class - XI
Quadratic Equations
Number
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?
Class - XI
Quadratic Equations
Topic: Quadratic Equations

What is Quadratic
Equation?
Class - XI
Quadratic Equations
2
An 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 com plex)
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 D
x
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 D
x 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.
Class - XI
Quadratic Equations
Important Facts

Fact # 2
0 is both purely real and purely imaginary number.
Class - XI
Quadratic Equations
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.
Class - XI
Quadratic Equations
Important Facts

Fact # 4
All purely imaginary number except zero are imaginary
numbers, but an imaginary number may or may not be purely
imaginary.
Class - XI
Quadratic Equations
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).
Class - XI
Quadratic Equations
Important Facts

Fact # 6
Class - XI
Quadratic Equations
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 num berz 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 com m utative 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 ibThe existence of additive inverse : we have the complex number
(denoted as ), called the additive inverse or negativea i b z
of , such that 0.z z z

Fact # 7
Class - XI
Quadratic Equations
Important Facts
1 2
(a) The product of two complex numbers is a complex number, the product is a complex numberz z
The m ultiplication of com plex num bers satisfy the follow ing 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 com m utative 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
iThe existence of m ultiplicative 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 ibThe existence of m ultiplicative inverse :
1
2 2 2 2
1
denoted by or called the multiplicative inverse of
a b
z 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) .
z
z
z 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.
Class - XI
Quadratic Equations
Important Facts
Number

Fact # 9
The fundamental theorem of algebra states that every non-
constant single-variable polynomials with complex coefficients has
at least one complex root.
Class - XI
Quadratic Equations
Important Facts

Fact # 10
Every polynomial equation of degree n has n roots.
Class - XI
Quadratic Equations
Important Facts

Fact # 11
Class - XI
Quadratic Equations
Important Facts
2
If 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

Class - XI
Quadratic Equations
Important Formulae
1. iIn te g ra l p o w e rs o f :
2
3 2 4 2 2
2 2
2 1 2
W e have 1, 1. T herefore,
1 , 1 1 1.
W e note that for any
1, w hen is even
(a) 1 .
1, w hen is odd
, w hen is even
(b) 1
, w hen i
n nn
nn n
i i
i i i i i i i i
n N
n
i i
n
i n
i i i i
i n
.
s odd
1
Also, for any , the value of is found out by w riting this as and solving .
T hus, any integral pow er of can be expressed in term s of 1 or .
n n
n
n N i i
i
i i

Class - XI
Quadratic Equations
Important Formulae
2. Identities for com plex num bers :
1 2
2 2 2
1 2 1 2 1 2
2 2 2
1 2 1 1 2 2
3 3 2 2 3
1 2 1 1 2 1 2 2
3 3 2 2 3
1 2 1 1 2 1 2 2
2 2
1 2 1 2 1 2
For all com plex num bers and w e get the follow ing:
(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

Class - XI
Quadratic Equations
Important Formulae
3. P roperties of conjugate :
2 2
(a) ; (b) 2 R e ;
(c) 2 Im ; (d) R e 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 1
2
2 2
(f) if and only if is purely im aginary;
(g) ; (h) ;
(i) , w here 0.
z z z
z z z z z z z z
z z
z
z z

Class - XI
Quadratic Equations
Important Formulae
4. P roperties of m odulus of a com plex num ber :
1 2
1 2 1 2
1 1
2
2 2
For any tw o com plex num ber and , w e hav e
(a)
(b) provided 0.
z z
z z z z
z z
z
z z

Method to find the multiplicative inverse of a non zero
complex number x + iy:
Class – XI,CBSE
Important Procedures
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 1
M ultiplicative inverse of
.
x iy x iy x iy
x iy
x iy x iy x iy x i y x y
x y
i
x y x
Quadratic Equations

Method to write a complex number in the form A + iB:
Class – XI,CBSE
Quadratic Equations
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
W e have
, where and .
a ib c ida ib
c id c id c id
ac bd i bc ad ac bd bc ad
i
c d c d c d
ac bd bc ad
A iB A B
c d c d
Quadratic Equations
a ib
c id

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