NCERT Textbook: Complex Numbers & Quadratic Equations | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 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
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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
? ?
- -
? ? +
? ?
+ - + -
? ?
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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
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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 =
.
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