Page 1
76 MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(18051865)
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 nonnegative. 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
202425
Page 2
76 MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(18051865)
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 nonnegative. 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
202425
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
202425
Page 3
76 MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(18051865)
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 nonnegative. 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
202425
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
202425
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 nonzero 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
? ?
 
? ? +
? ?
+  + 
? ?
202425
Page 4
76 MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(18051865)
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 nonnegative. 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
202425
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
202425
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 nonzero 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
? ?
 
? ? +
? ?
+  + 
? ?
202425
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
202425
Page 5
76 MATHEMATICS
Chapter
COMPLEX NUMBERS AND
QUADRATIC EQUATIONS
W. R. Hamilton
(18051865)
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 nonnegative. 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
202425
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 nonzero 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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