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 Page 1


76       MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(1805-1865)
vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v
4.1  Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
 + 1 = 0 has no real
solution as x
2
 + 1 = 0 gives x
2
 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x
2
 = – 1. In fact, the main
objective is to solve the equation ax
2
 + bx + c = 0, where
D = b
2
 –  4ac < 0, which is not possible in the system of
real numbers.
4.2  Complex Numbers
Let us denote 
1 -
 by the symbol i. Then, we have 
2
1 i = - . This means that i is a
solution of the equation x
2
 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
complex number. For example, 2 + i3,  (– 1) + 
3 i
,  
1
4
11
i
- ? ?
+
? ?
? ?
 are complex numbers.
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z
1
 = a + ib and z
2
 = c + id are equal if  a = c and b = d.
4
2024-25
Page 2


76       MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(1805-1865)
vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v
4.1  Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
 + 1 = 0 has no real
solution as x
2
 + 1 = 0 gives x
2
 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x
2
 = – 1. In fact, the main
objective is to solve the equation ax
2
 + bx + c = 0, where
D = b
2
 –  4ac < 0, which is not possible in the system of
real numbers.
4.2  Complex Numbers
Let us denote 
1 -
 by the symbol i. Then, we have 
2
1 i = - . This means that i is a
solution of the equation x
2
 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
complex number. For example, 2 + i3,  (– 1) + 
3 i
,  
1
4
11
i
- ? ?
+
? ?
? ?
 are complex numbers.
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z
1
 = a + ib and z
2
 = c + id are equal if  a = c and b = d.
4
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       77
Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find
the values of x and y.
Solution We have
4x + i (3x – y) = 3 + i (–6) ... (1)
Equating the real and the imaginary parts of (1), we get
4x = 3, 3x – y = – 6,
which, on solving simultaneously, give  
3
4
x=
 and 
33
4
y =
.
4.3   Algebra of Complex Numbers
In this Section, we shall develop the algebra of complex numbers.
4.3.1   Addition of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any two
complex numbers. Then, the sum  z
1
 + z
2
 is defined as follows:
z
1
 + z
2
 = (a + c) + i (b + d), which is again a complex number.
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties:
(i) The closure law  The sum of two complex numbers is a complex
number, i.e., z
1
 + z
2
 is a complex number for all complex numbers
z
1
 and z
2
.
(ii) The commutative law  For any two complex numbers z
1
 and z
2
,
z
1
 + z
2
 = z
2
 
+ z
1
(iii) The associative law  For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 + z
2
) + z
3
 = z
1
 + (z
2
 + z
3
).
(iv) The existence of additive identity  There exists the complex number
0 + i 0 (denoted as 0), called the additive identity or the zero complex
number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse  To every complex number
z = a + ib, we have the complex number – a + i(– b) (denoted as –  z),
called the additive inverse or negative of z. We observe that z + (–z) = 0
(the additive identity).
4.3.2  Difference of two complex numbers Given any two complex numbers z
1
 and
z
2
, the difference z
1
 – z
2
 is defined as follows:
z
1
 – z
2
 = z
1
 + (– z
2
).
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i
and (2 –  i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i
2024-25
Page 3


76       MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(1805-1865)
vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v
4.1  Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
 + 1 = 0 has no real
solution as x
2
 + 1 = 0 gives x
2
 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x
2
 = – 1. In fact, the main
objective is to solve the equation ax
2
 + bx + c = 0, where
D = b
2
 –  4ac < 0, which is not possible in the system of
real numbers.
4.2  Complex Numbers
Let us denote 
1 -
 by the symbol i. Then, we have 
2
1 i = - . This means that i is a
solution of the equation x
2
 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
complex number. For example, 2 + i3,  (– 1) + 
3 i
,  
1
4
11
i
- ? ?
+
? ?
? ?
 are complex numbers.
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z
1
 = a + ib and z
2
 = c + id are equal if  a = c and b = d.
4
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       77
Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find
the values of x and y.
Solution We have
4x + i (3x – y) = 3 + i (–6) ... (1)
Equating the real and the imaginary parts of (1), we get
4x = 3, 3x – y = – 6,
which, on solving simultaneously, give  
3
4
x=
 and 
33
4
y =
.
4.3   Algebra of Complex Numbers
In this Section, we shall develop the algebra of complex numbers.
4.3.1   Addition of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any two
complex numbers. Then, the sum  z
1
 + z
2
 is defined as follows:
z
1
 + z
2
 = (a + c) + i (b + d), which is again a complex number.
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties:
(i) The closure law  The sum of two complex numbers is a complex
number, i.e., z
1
 + z
2
 is a complex number for all complex numbers
z
1
 and z
2
.
(ii) The commutative law  For any two complex numbers z
1
 and z
2
,
z
1
 + z
2
 = z
2
 
+ z
1
(iii) The associative law  For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 + z
2
) + z
3
 = z
1
 + (z
2
 + z
3
).
(iv) The existence of additive identity  There exists the complex number
0 + i 0 (denoted as 0), called the additive identity or the zero complex
number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse  To every complex number
z = a + ib, we have the complex number – a + i(– b) (denoted as –  z),
called the additive inverse or negative of z. We observe that z + (–z) = 0
(the additive identity).
4.3.2  Difference of two complex numbers Given any two complex numbers z
1
 and
z
2
, the difference z
1
 – z
2
 is defined as follows:
z
1
 – z
2
 = z
1
 + (– z
2
).
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i
and (2 –  i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i
2024-25
78       MATHEMATICS
4.3.3  Multiplication of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any
two complex numbers. Then, the product z
1
 z
2
 is defined as follows:
z
1
 z
2
 = (ac –  bd) + i(ad + bc)
For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28
The multiplication of complex numbers possesses the following properties, which
we state without proofs.
(i) The closure law The product of two complex numbers is a complex number,
the product z
1
 z
2
 is a complex number for all complex numbers z
1
 and z
2
.
(ii) The commutative law For any two complex numbers z
1
 and z
2
,
z
1
 z
2
 = z
2
 z
1
.
(iii) The associative law For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 z
2
) z
3
 = z
1
 (z
2
 z
3
).
(iv) The existence of multiplicative identity There exists the complex number
1 + i 0 (denoted as 1),  called the multiplicative identity such that z.1 = z,
for every complex number z.
(v) The existence of multiplicative inverse For every non-zero complex
number z = a + ib or a + bi(a ? 0, b ? 0), we have the complex number
2 2 2 2
a –b
i
a b a b
+
+ +
 (denoted by 
1
z
 or z
–1
 ), called the multiplicative inverse
of z such that
1
1 z.
z
=
 (the multiplicative identity).
(vi) The distributive law For any three complex numbers z
1
, z
2
, z
3
,
(a)  z
1
 (z
2
 + z
3
) = z
1
 z
2
 + z
1
 z
3
(b)  (z
1
 + z
2
) z
3
 = z
1
 z
3
 + z
2
 z
3
4.3.4  Division of two complex numbers Given any two complex numbers z
1 
and  z
2
,
where 
2
0 z ? , the quotient 
1
2
z
z
 is defined by
1
1
2 2
1 z
z
z z
=
For example, let z
1
 = 6 + 3i and  z
2
 = 2 – i
Then
1
2
1
(6 3 )
2
z
i
z i
? ?
= + ×
? ?
-
? ?
 = ( ) 6 3i + 
( )
( )
( )
2 2
2 2
1 2
2 1 2 1
i
? ?
- -
? ? +
? ?
+ - + -
? ?
2024-25
Page 4


76       MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(1805-1865)
vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v
4.1  Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
 + 1 = 0 has no real
solution as x
2
 + 1 = 0 gives x
2
 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x
2
 = – 1. In fact, the main
objective is to solve the equation ax
2
 + bx + c = 0, where
D = b
2
 –  4ac < 0, which is not possible in the system of
real numbers.
4.2  Complex Numbers
Let us denote 
1 -
 by the symbol i. Then, we have 
2
1 i = - . This means that i is a
solution of the equation x
2
 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
complex number. For example, 2 + i3,  (– 1) + 
3 i
,  
1
4
11
i
- ? ?
+
? ?
? ?
 are complex numbers.
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z
1
 = a + ib and z
2
 = c + id are equal if  a = c and b = d.
4
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       77
Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find
the values of x and y.
Solution We have
4x + i (3x – y) = 3 + i (–6) ... (1)
Equating the real and the imaginary parts of (1), we get
4x = 3, 3x – y = – 6,
which, on solving simultaneously, give  
3
4
x=
 and 
33
4
y =
.
4.3   Algebra of Complex Numbers
In this Section, we shall develop the algebra of complex numbers.
4.3.1   Addition of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any two
complex numbers. Then, the sum  z
1
 + z
2
 is defined as follows:
z
1
 + z
2
 = (a + c) + i (b + d), which is again a complex number.
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties:
(i) The closure law  The sum of two complex numbers is a complex
number, i.e., z
1
 + z
2
 is a complex number for all complex numbers
z
1
 and z
2
.
(ii) The commutative law  For any two complex numbers z
1
 and z
2
,
z
1
 + z
2
 = z
2
 
+ z
1
(iii) The associative law  For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 + z
2
) + z
3
 = z
1
 + (z
2
 + z
3
).
(iv) The existence of additive identity  There exists the complex number
0 + i 0 (denoted as 0), called the additive identity or the zero complex
number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse  To every complex number
z = a + ib, we have the complex number – a + i(– b) (denoted as –  z),
called the additive inverse or negative of z. We observe that z + (–z) = 0
(the additive identity).
4.3.2  Difference of two complex numbers Given any two complex numbers z
1
 and
z
2
, the difference z
1
 – z
2
 is defined as follows:
z
1
 – z
2
 = z
1
 + (– z
2
).
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i
and (2 –  i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i
2024-25
78       MATHEMATICS
4.3.3  Multiplication of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any
two complex numbers. Then, the product z
1
 z
2
 is defined as follows:
z
1
 z
2
 = (ac –  bd) + i(ad + bc)
For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28
The multiplication of complex numbers possesses the following properties, which
we state without proofs.
(i) The closure law The product of two complex numbers is a complex number,
the product z
1
 z
2
 is a complex number for all complex numbers z
1
 and z
2
.
(ii) The commutative law For any two complex numbers z
1
 and z
2
,
z
1
 z
2
 = z
2
 z
1
.
(iii) The associative law For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 z
2
) z
3
 = z
1
 (z
2
 z
3
).
(iv) The existence of multiplicative identity There exists the complex number
1 + i 0 (denoted as 1),  called the multiplicative identity such that z.1 = z,
for every complex number z.
(v) The existence of multiplicative inverse For every non-zero complex
number z = a + ib or a + bi(a ? 0, b ? 0), we have the complex number
2 2 2 2
a –b
i
a b a b
+
+ +
 (denoted by 
1
z
 or z
–1
 ), called the multiplicative inverse
of z such that
1
1 z.
z
=
 (the multiplicative identity).
(vi) The distributive law For any three complex numbers z
1
, z
2
, z
3
,
(a)  z
1
 (z
2
 + z
3
) = z
1
 z
2
 + z
1
 z
3
(b)  (z
1
 + z
2
) z
3
 = z
1
 z
3
 + z
2
 z
3
4.3.4  Division of two complex numbers Given any two complex numbers z
1 
and  z
2
,
where 
2
0 z ? , the quotient 
1
2
z
z
 is defined by
1
1
2 2
1 z
z
z z
=
For example, let z
1
 = 6 + 3i and  z
2
 = 2 – i
Then
1
2
1
(6 3 )
2
z
i
z i
? ?
= + ×
? ?
-
? ?
 = ( ) 6 3i + 
( )
( )
( )
2 2
2 2
1 2
2 1 2 1
i
? ?
- -
? ? +
? ?
+ - + -
? ?
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       79
= 
( )
2
6 3
5
i
i
+ ? ?
+
? ?
? ?
 = ( ) ( )
1 1
12 3 6 6 9 12
5 5
i i ? - + + ? = +
? ?
4.3.5  Power of i  we know that
( )
3 2
1 i i i i i = = - = - ,
( )
( )
2
2
4 2
1 1 i i = = - =
( )
( )
2
2
5 2
1 i i i i i = = - = , 
( )
( )
3
3
6 2
1 1 i i = = - = - , etc.
Also, we have    
1 2
2
1 1 1
, 1,
1 1
i i
i i i
i i i
- -
= × = = - = = =-
- -
     
3 4
3 4
1 1 1 1
, 1
1 1
i i
i i i
i i i i
- -
= = × = = = = =
-
In general, for any integer k, i
4k
 = 1, i
4k + 1
 = i, i
4k + 2
 = –1, i
4k + 3
 = – i
4.3.6  The square roots of a negative real number
Note that  i
2
 = –1 and  ( – i)
2
 = i
2
 = – 1
Therefore,  the square roots of – 1 are i, – i. However, by the symbol 
1 -
, we would
mean i only.
Now, we can see that i and –i both are the solutions of the equation x
2
 + 1 = 0 or
x
2
 = –1.
Similarly
( ) ( )
2 2
3 3 i = i
2
 = 3 (– 1) = – 3
( )
2
3i - = 
( )
2
3 - i
2
 = – 3
Therefore,  the square roots of –3 are 3 i and 3i - .
Again, the symbol 
3 -
 is meant to represent 3 i only, i.e., 
3 -
 = 3 i .
Generally, if a is a positive real number, a - = 1 a - = a i ,
We already know that a b × = 
ab
 for all positive real number a and b. This
result also holds true when either a > 0, b < 0  or a < 0,  b > 0. What if a < 0, b < 0?
Let us examine.
Note that
2024-25
Page 5


76       MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(1805-1865)
vMathematics is the Queen of Sciences and Arithmetic is the Queen of
Mathematics. – GAUSS v
4.1  Introduction
In earlier classes, we have studied linear equations in one
and two variables and quadratic equations in one variable.
We have seen that the equation x
2
 + 1 = 0 has no real
solution as x
2
 + 1 = 0 gives x
2
 = – 1 and square of every
real number is non-negative. So, we need to extend the
real number system to a larger system so that we can
find the solution of the equation x
2
 = – 1. In fact, the main
objective is to solve the equation ax
2
 + bx + c = 0, where
D = b
2
 –  4ac < 0, which is not possible in the system of
real numbers.
4.2  Complex Numbers
Let us denote 
1 -
 by the symbol i. Then, we have 
2
1 i = - . This means that i is a
solution of the equation x
2
 + 1 = 0.
A number of the form a + ib, where a and b are real numbers, is defined to be a
complex number. For example, 2 + i3,  (– 1) + 
3 i
,  
1
4
11
i
- ? ?
+
? ?
? ?
 are complex numbers.
For the complex number z = a + ib, a is called the real part, denoted by Re z and
b is called the imaginary part denoted by Im z of the complex number z. For example,
if z = 2 + i5, then Re z = 2 and Im z = 5.
Two complex numbers z
1
 = a + ib and z
2
 = c + id are equal if  a = c and b = d.
4
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       77
Example 1 If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find
the values of x and y.
Solution We have
4x + i (3x – y) = 3 + i (–6) ... (1)
Equating the real and the imaginary parts of (1), we get
4x = 3, 3x – y = – 6,
which, on solving simultaneously, give  
3
4
x=
 and 
33
4
y =
.
4.3   Algebra of Complex Numbers
In this Section, we shall develop the algebra of complex numbers.
4.3.1   Addition of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any two
complex numbers. Then, the sum  z
1
 + z
2
 is defined as follows:
z
1
 + z
2
 = (a + c) + i (b + d), which is again a complex number.
For example, (2 + i3) + (– 6 +i5) = (2 – 6) + i (3 + 5) = – 4 + i 8
The addition of complex numbers satisfy the following properties:
(i) The closure law  The sum of two complex numbers is a complex
number, i.e., z
1
 + z
2
 is a complex number for all complex numbers
z
1
 and z
2
.
(ii) The commutative law  For any two complex numbers z
1
 and z
2
,
z
1
 + z
2
 = z
2
 
+ z
1
(iii) The associative law  For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 + z
2
) + z
3
 = z
1
 + (z
2
 + z
3
).
(iv) The existence of additive identity  There exists the complex number
0 + i 0 (denoted as 0), called the additive identity or the zero complex
number, such that, for every complex number z, z + 0 = z.
(v) The existence of additive inverse  To every complex number
z = a + ib, we have the complex number – a + i(– b) (denoted as –  z),
called the additive inverse or negative of z. We observe that z + (–z) = 0
(the additive identity).
4.3.2  Difference of two complex numbers Given any two complex numbers z
1
 and
z
2
, the difference z
1
 – z
2
 is defined as follows:
z
1
 – z
2
 = z
1
 + (– z
2
).
For example, (6 + 3i) – (2 – i) = (6 + 3i) + (– 2 + i ) = 4 + 4i
and (2 –  i) – (6 + 3i) = (2 – i) + ( – 6 – 3i) = – 4 – 4i
2024-25
78       MATHEMATICS
4.3.3  Multiplication of two complex numbers Let z
1
 = a + ib and z
2
 = c + id be any
two complex numbers. Then, the product z
1
 z
2
 is defined as follows:
z
1
 z
2
 = (ac –  bd) + i(ad + bc)
For example, (3 + i5) (2 + i6) = (3 × 2 – 5 × 6) + i(3 × 6 + 5 × 2) = – 24 + i28
The multiplication of complex numbers possesses the following properties, which
we state without proofs.
(i) The closure law The product of two complex numbers is a complex number,
the product z
1
 z
2
 is a complex number for all complex numbers z
1
 and z
2
.
(ii) The commutative law For any two complex numbers z
1
 and z
2
,
z
1
 z
2
 = z
2
 z
1
.
(iii) The associative law For any three complex numbers z
1
, z
2
, z
3
,
(z
1
 z
2
) z
3
 = z
1
 (z
2
 z
3
).
(iv) The existence of multiplicative identity There exists the complex number
1 + i 0 (denoted as 1),  called the multiplicative identity such that z.1 = z,
for every complex number z.
(v) The existence of multiplicative inverse For every non-zero complex
number z = a + ib or a + bi(a ? 0, b ? 0), we have the complex number
2 2 2 2
a –b
i
a b a b
+
+ +
 (denoted by 
1
z
 or z
–1
 ), called the multiplicative inverse
of z such that
1
1 z.
z
=
 (the multiplicative identity).
(vi) The distributive law For any three complex numbers z
1
, z
2
, z
3
,
(a)  z
1
 (z
2
 + z
3
) = z
1
 z
2
 + z
1
 z
3
(b)  (z
1
 + z
2
) z
3
 = z
1
 z
3
 + z
2
 z
3
4.3.4  Division of two complex numbers Given any two complex numbers z
1 
and  z
2
,
where 
2
0 z ? , the quotient 
1
2
z
z
 is defined by
1
1
2 2
1 z
z
z z
=
For example, let z
1
 = 6 + 3i and  z
2
 = 2 – i
Then
1
2
1
(6 3 )
2
z
i
z i
? ?
= + ×
? ?
-
? ?
 = ( ) 6 3i + 
( )
( )
( )
2 2
2 2
1 2
2 1 2 1
i
? ?
- -
? ? +
? ?
+ - + -
? ?
2024-25
COMPLEX NUMBERS AND QUADRATIC EQUATIONS       79
= 
( )
2
6 3
5
i
i
+ ? ?
+
? ?
? ?
 = ( ) ( )
1 1
12 3 6 6 9 12
5 5
i i ? - + + ? = +
? ?
4.3.5  Power of i  we know that
( )
3 2
1 i i i i i = = - = - ,
( )
( )
2
2
4 2
1 1 i i = = - =
( )
( )
2
2
5 2
1 i i i i i = = - = , 
( )
( )
3
3
6 2
1 1 i i = = - = - , etc.
Also, we have    
1 2
2
1 1 1
, 1,
1 1
i i
i i i
i i i
- -
= × = = - = = =-
- -
     
3 4
3 4
1 1 1 1
, 1
1 1
i i
i i i
i i i i
- -
= = × = = = = =
-
In general, for any integer k, i
4k
 = 1, i
4k + 1
 = i, i
4k + 2
 = –1, i
4k + 3
 = – i
4.3.6  The square roots of a negative real number
Note that  i
2
 = –1 and  ( – i)
2
 = i
2
 = – 1
Therefore,  the square roots of – 1 are i, – i. However, by the symbol 
1 -
, we would
mean i only.
Now, we can see that i and –i both are the solutions of the equation x
2
 + 1 = 0 or
x
2
 = –1.
Similarly
( ) ( )
2 2
3 3 i = i
2
 = 3 (– 1) = – 3
( )
2
3i - = 
( )
2
3 - i
2
 = – 3
Therefore,  the square roots of –3 are 3 i and 3i - .
Again, the symbol 
3 -
 is meant to represent 3 i only, i.e., 
3 -
 = 3 i .
Generally, if a is a positive real number, a - = 1 a - = a i ,
We already know that a b × = 
ab
 for all positive real number a and b. This
result also holds true when either a > 0, b < 0  or a < 0,  b > 0. What if a < 0, b < 0?
Let us examine.
Note that
2024-25
80       MATHEMATICS
( ) ( )
2
1 1 1 1 i = - - = - - (by assuming 
a b ×
 = 
ab
 for all real numbers)
= 1 = 1,  which is a contradiction to the fact that = -
2
1 i .
Therefore, 
a b ab × ?
 if both a and b are negative real numbers.
Further, if any of a and b is zero, then, clearly, 
a b ab × =
= 0.
4.3.7 Identities We prove the following identity
( )
2
2 2
1 2 1 2 1 2
2 z z z z z z + = + + , for all complex numbers z
1
 and z
2
.
Proof  We have, (z
1
 + z
2
)
2
 = (z
1
 + z
2
) (z
1
 + z
2
),
=  (z
1
 + z
2
) z
1
 + (z
1
 + z
2
) z
2
(Distributive law)
= 
2 2
1 2 1 1 2 2
z z z z z z + + + (Distributive law)
    = 
2 2
1 1 2 1 2 2
z z z z z z + + + (Commutative law of multiplication)
    = 
2 2
1 1 2 2
2 z z z z + +
Similarly, we can prove the following identities:
(i)
( )
2
2 2
1 2 1 1 2 2
2 z z z z z z - = - +
(ii)
( )
3
3 2 2 3
1 2 1 1 2 1 2 2
3 3 z z z z z z z z + = + + +
(iii)
( )
3
3 2 2 3
1 2 1 1 2 1 2 2
3 3 z z z z z z z z - = - + -
(iv)
( )( )
2 2
1 2 1 2 1 2
z – z z z z – z = +
In fact, many other identities which are true for all real numbers, can be proved
to be true for all complex numbers.
Example 2 Express the following in the form of  a + bi:
(i)
( )
1
5
8
i i
? ?
-
? ?
? ?
(ii) ( ) ( ) 2 i i - 
3
1
8
i
? ?
-
? ?
? ?
Solution (i)
( )
1
5
8
i i
? ?
-
? ?
? ?
 = 
2
5
8
i
-
 = ( )
5
1
8
-
-
 = 
5
8
= 
5
0
8
i +
(ii) ( ) ( )
3
1
2
8
i i i
? ?
- -
? ?
? ?
 = 
5
1
2
8 8 8
i × ×
× ×
  = ( )
2
2
1
256
i
 
1
256
i i =
.
2024-25
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FAQs on NCERT Textbook: Complex Numbers & Quadratic Equations - NCERT Textbooks (Class 6 to Class 12) - CTET & State TET

1. What are complex numbers?
Ans. Complex numbers are numbers that can be expressed in the form a + bi, where "a" and "b" are real numbers, and "i" is the imaginary unit defined as the square root of -1. These numbers consist of a real part and an imaginary part.
2. How are complex numbers represented on the complex plane?
Ans. Complex numbers can be represented on the complex plane, also known as the Argand plane. The real part of the complex number is plotted on the horizontal axis, while the imaginary part is plotted on the vertical axis. The point representing the complex number is then located at the intersection of these two axes.
3. What is the conjugate of a complex number?
Ans. The conjugate of a complex number a + bi is obtained by changing the sign of the imaginary part. In other words, the conjugate of a + bi is a - bi. For example, the conjugate of 3 + 2i is 3 - 2i.
4. How do we add and subtract complex numbers?
Ans. To add or subtract complex numbers, we simply add or subtract their real parts separately and their imaginary parts separately. For example, to add (3 + 2i) and (1 - 4i), we add 3 + 1 to get 4 and 2i - 4i to get -2i. Therefore, the sum is 4 - 2i.
5. How do we multiply and divide complex numbers?
Ans. To multiply complex numbers, we use the distributive property and simplify the resulting expression. To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator, simplifying the expression. Both multiplication and division of complex numbers involve multiplying and dividing their real and imaginary parts separately.
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