Detailed Notes: Complex Numbers | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Complex Numbers 
Complex numbers, an extension of the real numbers, are fundamental in various branches of 
mathematics, science, and engineering. Introduced to solve equations that do not have real 
solutions, complex numbers have since become a critical tool in numerous theoretical and 
practical applications. They originated from the need to find solutions to quadratic equations 
where the discriminant (the term under the square root) was negative, thereby indicating no 
real roots. This necessity led mathematicians to define the imaginary unit iii, where i2=-1i^2 = -
1i2=-1, allowing for the creation of numbers of the form a+bia + bia+bi, where a and b are real 
numbers. Complex numbers are now indispensable in fields such as electrical engineering, 
quantum mechanics, and signal processing, demonstrating their profound utility beyond their 
theoretical origins. 
The complex number system 
A complex number (?? ) is a number that can be expressed in the form ?? = ?? + ???? where ?? 
and ?? are real numbers and ?? 2
= -1. Here  'a' is called as real part of ?? which is denoted 
by (???? ??? ) and ' ?? ' is called as imaginary part of ?? , which is denoted by (Im z). 
Any complex number is : 
(i) Purely real, if ?? = 0 
(ii) Imaginary, if ?? ? 0. 
(iii) Purely imaginary, if ?? = 0 
Note : 
(a) The set ?? of real numbers is a proper subset of the Complex Numbers. Hence the 
complete number system is ?? ? ?? ? ?? ? ?? ? ?? ? ?? . 
(b) Zero is purely real as well as purely imaginary but not imaginary. 
(c) ?? = v-1 is called the imaginary unit. 
 ???????? ?? 2
= -1;?? 3
= -?? ;?? 4
= 1 ?????? .  
(d) v?? v?? = v???? only if atleast one of ?? or ?? is non-negative. 
(e) If ?? = ?? + ???? , then ?? - ???? is called complex conjugate of ?? and written as ?? ? = ?? - ???? 
(f) Real numbers satisfy order relations, where as imaginary numbers do not satisfy 
order relations i.e. ??? > 0,3+ ?? < 2 are meaningless. 
Page 2


Complex Numbers 
Complex numbers, an extension of the real numbers, are fundamental in various branches of 
mathematics, science, and engineering. Introduced to solve equations that do not have real 
solutions, complex numbers have since become a critical tool in numerous theoretical and 
practical applications. They originated from the need to find solutions to quadratic equations 
where the discriminant (the term under the square root) was negative, thereby indicating no 
real roots. This necessity led mathematicians to define the imaginary unit iii, where i2=-1i^2 = -
1i2=-1, allowing for the creation of numbers of the form a+bia + bia+bi, where a and b are real 
numbers. Complex numbers are now indispensable in fields such as electrical engineering, 
quantum mechanics, and signal processing, demonstrating their profound utility beyond their 
theoretical origins. 
The complex number system 
A complex number (?? ) is a number that can be expressed in the form ?? = ?? + ???? where ?? 
and ?? are real numbers and ?? 2
= -1. Here  'a' is called as real part of ?? which is denoted 
by (???? ??? ) and ' ?? ' is called as imaginary part of ?? , which is denoted by (Im z). 
Any complex number is : 
(i) Purely real, if ?? = 0 
(ii) Imaginary, if ?? ? 0. 
(iii) Purely imaginary, if ?? = 0 
Note : 
(a) The set ?? of real numbers is a proper subset of the Complex Numbers. Hence the 
complete number system is ?? ? ?? ? ?? ? ?? ? ?? ? ?? . 
(b) Zero is purely real as well as purely imaginary but not imaginary. 
(c) ?? = v-1 is called the imaginary unit. 
 ???????? ?? 2
= -1;?? 3
= -?? ;?? 4
= 1 ?????? .  
(d) v?? v?? = v???? only if atleast one of ?? or ?? is non-negative. 
(e) If ?? = ?? + ???? , then ?? - ???? is called complex conjugate of ?? and written as ?? ? = ?? - ???? 
(f) Real numbers satisfy order relations, where as imaginary numbers do not satisfy 
order relations i.e. ??? > 0,3+ ?? < 2 are meaningless. 
Fundamental operations with complex 
numbers 
In performing operations with complex numbers we can proceed as in the algebra of real 
numbers, replacing ?? 2
 by -1 when it occurs. 
1. Addition (?? + ???? )+ (?? + ???? ) = ?? + ???? + ?? + ???? = (?? + ?? )+ (?? + ?? )?? 
2. Subtraction (?? + ???? )- (?? + ???? )= ?? + ???? - ?? - ???? = (?? - ?? )+ (?? - ?? )?? 
3. Multiplication (?? + ???? )(?? + ???? )= ???? + ?????? + ?????? + ???? ?? 2
= (???? - ???? )+ (???? + ???? )?? 
4. Division 
?? +????
?? +????
=
?? +????
?? +????
·
?? -????
?? -????
=
???? -?????? +?????? -???? ?? 2
?? 2
-?? 2
?? 2
=
???? +???? +(???? -???? )?? ?? 2
+?? 2
=
???? +????
?? 2
+?? 2
+
???? -????
?? 2
+?? 2
?? 
Inequalities in imaginary numbers are not defined. There is no validity if we say that 
imaginary number is positive or negative. 
e.g. ??? > 0,4+ 2?? < 2 + 4 i are meaningless. 
In real numbers if ?? 2
+ ?? 2
= 0 then ?? = 0 = ?? however in complex numbers, 
?? 1
2
+ ?? 2
2
= 0 ???????? ??????? ??????????? ?? 1
= ?? 2
= 0 
Problem 1 : Find the multiplicative inverse of 4+ 3?? . 
Solution : Let ?? be the multiplicative inverse of 4 + 3?? then 
?? (4 + 3?? )= 1 
?? =
1
4+ 3?? ×
4 - 3?? 4 - 3?? =
4 - 3?? 16+ 9
=
4- 3?? 25
? ?????? . ?
4 - 3?? 25
 
Equality In Complex Number : 
Two complex numbers ?? 1
= ?? 1
+ ?? ?? 1
&?? 2
= ?? 2
+ ?? ?? 2
 are equal if and only if their real and 
imaginary parts are equal respectively 
i.e. ??? 1
= ?? 2
 
? ????? ?(?? 1
) = ???? ?(?? 2
) and ?? ?? (?? 1
) = ?? ?? (?? 2
). 
Problem 2 : Find the value of ?? and ?? for which 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3- 5?? )+ (1+ 2???? ), where ?? ,???? 
Page 3


Complex Numbers 
Complex numbers, an extension of the real numbers, are fundamental in various branches of 
mathematics, science, and engineering. Introduced to solve equations that do not have real 
solutions, complex numbers have since become a critical tool in numerous theoretical and 
practical applications. They originated from the need to find solutions to quadratic equations 
where the discriminant (the term under the square root) was negative, thereby indicating no 
real roots. This necessity led mathematicians to define the imaginary unit iii, where i2=-1i^2 = -
1i2=-1, allowing for the creation of numbers of the form a+bia + bia+bi, where a and b are real 
numbers. Complex numbers are now indispensable in fields such as electrical engineering, 
quantum mechanics, and signal processing, demonstrating their profound utility beyond their 
theoretical origins. 
The complex number system 
A complex number (?? ) is a number that can be expressed in the form ?? = ?? + ???? where ?? 
and ?? are real numbers and ?? 2
= -1. Here  'a' is called as real part of ?? which is denoted 
by (???? ??? ) and ' ?? ' is called as imaginary part of ?? , which is denoted by (Im z). 
Any complex number is : 
(i) Purely real, if ?? = 0 
(ii) Imaginary, if ?? ? 0. 
(iii) Purely imaginary, if ?? = 0 
Note : 
(a) The set ?? of real numbers is a proper subset of the Complex Numbers. Hence the 
complete number system is ?? ? ?? ? ?? ? ?? ? ?? ? ?? . 
(b) Zero is purely real as well as purely imaginary but not imaginary. 
(c) ?? = v-1 is called the imaginary unit. 
 ???????? ?? 2
= -1;?? 3
= -?? ;?? 4
= 1 ?????? .  
(d) v?? v?? = v???? only if atleast one of ?? or ?? is non-negative. 
(e) If ?? = ?? + ???? , then ?? - ???? is called complex conjugate of ?? and written as ?? ? = ?? - ???? 
(f) Real numbers satisfy order relations, where as imaginary numbers do not satisfy 
order relations i.e. ??? > 0,3+ ?? < 2 are meaningless. 
Fundamental operations with complex 
numbers 
In performing operations with complex numbers we can proceed as in the algebra of real 
numbers, replacing ?? 2
 by -1 when it occurs. 
1. Addition (?? + ???? )+ (?? + ???? ) = ?? + ???? + ?? + ???? = (?? + ?? )+ (?? + ?? )?? 
2. Subtraction (?? + ???? )- (?? + ???? )= ?? + ???? - ?? - ???? = (?? - ?? )+ (?? - ?? )?? 
3. Multiplication (?? + ???? )(?? + ???? )= ???? + ?????? + ?????? + ???? ?? 2
= (???? - ???? )+ (???? + ???? )?? 
4. Division 
?? +????
?? +????
=
?? +????
?? +????
·
?? -????
?? -????
=
???? -?????? +?????? -???? ?? 2
?? 2
-?? 2
?? 2
=
???? +???? +(???? -???? )?? ?? 2
+?? 2
=
???? +????
?? 2
+?? 2
+
???? -????
?? 2
+?? 2
?? 
Inequalities in imaginary numbers are not defined. There is no validity if we say that 
imaginary number is positive or negative. 
e.g. ??? > 0,4+ 2?? < 2 + 4 i are meaningless. 
In real numbers if ?? 2
+ ?? 2
= 0 then ?? = 0 = ?? however in complex numbers, 
?? 1
2
+ ?? 2
2
= 0 ???????? ??????? ??????????? ?? 1
= ?? 2
= 0 
Problem 1 : Find the multiplicative inverse of 4+ 3?? . 
Solution : Let ?? be the multiplicative inverse of 4 + 3?? then 
?? (4 + 3?? )= 1 
?? =
1
4+ 3?? ×
4 - 3?? 4 - 3?? =
4 - 3?? 16+ 9
=
4- 3?? 25
? ?????? . ?
4 - 3?? 25
 
Equality In Complex Number : 
Two complex numbers ?? 1
= ?? 1
+ ?? ?? 1
&?? 2
= ?? 2
+ ?? ?? 2
 are equal if and only if their real and 
imaginary parts are equal respectively 
i.e. ??? 1
= ?? 2
 
? ????? ?(?? 1
) = ???? ?(?? 2
) and ?? ?? (?? 1
) = ?? ?? (?? 2
). 
Problem 2 : Find the value of ?? and ?? for which 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3- 5?? )+ (1+ 2???? ), where ?? ,???? 
Solution : 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3 - 5?? )+ (1 + 2???? ) 
???? 4
- 3?? 2
- 4 = 0 ? ?? 2
= 4 ? ?? = ±2 
and ?2?? - ?? = -5+ 2?? 
2?? + 5 = 3?? 
when ?? = 2 ? ?? = 3 
and ?? = -2 ? ??? = 1/3? Ans. (2,3) or (-2,1/3) 
Problem 3 : Find the value of expression ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3, when ?? = -1 + ?? . 
Solution : ??? = -1 + ?? 
(?? + 1)
2
= ?? 2
 
?? 2
+ 2?? + 2 = 0 
now, ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3 = (?? 2
+ 2?? + 2)(?? 2
+ 2?? - 1)+ 5 = 5 
Problem 4 : Find the square root of -21- 20 
Solution : ? Let ?? + ???? = v-21- 20?? 
??(?? + ???? )
2
= -21- 20?? ???? 2
- ?? 2
= -21?????? = -10?? ???????? ?(?? )?\&?(???? ) ???? 2
= 4 ? ??? = ±2?? ?? h???? ?? = 2,?? = -5 ?????? ?? = -2,?? = 5???? + ????
= (2- ?? 5) ???? (-2+ ?? 5)? 
Representation of a complex number : 
To each complex number there corresponds one and only one point in plane, and 
conversely to each point in the plane there corresponds one and only one complex 
number. Because of this we often refer to the complex number ?? as the point ?? . 
(a) Cartesian Form (Geometric Representation): 
Every complex number ?? = ?? + ?? y can be represented by a point on the Cartesian plane 
known as complex plane (Argand diagram) by the ordered pair ( ?? ,?? ). 
Page 4


Complex Numbers 
Complex numbers, an extension of the real numbers, are fundamental in various branches of 
mathematics, science, and engineering. Introduced to solve equations that do not have real 
solutions, complex numbers have since become a critical tool in numerous theoretical and 
practical applications. They originated from the need to find solutions to quadratic equations 
where the discriminant (the term under the square root) was negative, thereby indicating no 
real roots. This necessity led mathematicians to define the imaginary unit iii, where i2=-1i^2 = -
1i2=-1, allowing for the creation of numbers of the form a+bia + bia+bi, where a and b are real 
numbers. Complex numbers are now indispensable in fields such as electrical engineering, 
quantum mechanics, and signal processing, demonstrating their profound utility beyond their 
theoretical origins. 
The complex number system 
A complex number (?? ) is a number that can be expressed in the form ?? = ?? + ???? where ?? 
and ?? are real numbers and ?? 2
= -1. Here  'a' is called as real part of ?? which is denoted 
by (???? ??? ) and ' ?? ' is called as imaginary part of ?? , which is denoted by (Im z). 
Any complex number is : 
(i) Purely real, if ?? = 0 
(ii) Imaginary, if ?? ? 0. 
(iii) Purely imaginary, if ?? = 0 
Note : 
(a) The set ?? of real numbers is a proper subset of the Complex Numbers. Hence the 
complete number system is ?? ? ?? ? ?? ? ?? ? ?? ? ?? . 
(b) Zero is purely real as well as purely imaginary but not imaginary. 
(c) ?? = v-1 is called the imaginary unit. 
 ???????? ?? 2
= -1;?? 3
= -?? ;?? 4
= 1 ?????? .  
(d) v?? v?? = v???? only if atleast one of ?? or ?? is non-negative. 
(e) If ?? = ?? + ???? , then ?? - ???? is called complex conjugate of ?? and written as ?? ? = ?? - ???? 
(f) Real numbers satisfy order relations, where as imaginary numbers do not satisfy 
order relations i.e. ??? > 0,3+ ?? < 2 are meaningless. 
Fundamental operations with complex 
numbers 
In performing operations with complex numbers we can proceed as in the algebra of real 
numbers, replacing ?? 2
 by -1 when it occurs. 
1. Addition (?? + ???? )+ (?? + ???? ) = ?? + ???? + ?? + ???? = (?? + ?? )+ (?? + ?? )?? 
2. Subtraction (?? + ???? )- (?? + ???? )= ?? + ???? - ?? - ???? = (?? - ?? )+ (?? - ?? )?? 
3. Multiplication (?? + ???? )(?? + ???? )= ???? + ?????? + ?????? + ???? ?? 2
= (???? - ???? )+ (???? + ???? )?? 
4. Division 
?? +????
?? +????
=
?? +????
?? +????
·
?? -????
?? -????
=
???? -?????? +?????? -???? ?? 2
?? 2
-?? 2
?? 2
=
???? +???? +(???? -???? )?? ?? 2
+?? 2
=
???? +????
?? 2
+?? 2
+
???? -????
?? 2
+?? 2
?? 
Inequalities in imaginary numbers are not defined. There is no validity if we say that 
imaginary number is positive or negative. 
e.g. ??? > 0,4+ 2?? < 2 + 4 i are meaningless. 
In real numbers if ?? 2
+ ?? 2
= 0 then ?? = 0 = ?? however in complex numbers, 
?? 1
2
+ ?? 2
2
= 0 ???????? ??????? ??????????? ?? 1
= ?? 2
= 0 
Problem 1 : Find the multiplicative inverse of 4+ 3?? . 
Solution : Let ?? be the multiplicative inverse of 4 + 3?? then 
?? (4 + 3?? )= 1 
?? =
1
4+ 3?? ×
4 - 3?? 4 - 3?? =
4 - 3?? 16+ 9
=
4- 3?? 25
? ?????? . ?
4 - 3?? 25
 
Equality In Complex Number : 
Two complex numbers ?? 1
= ?? 1
+ ?? ?? 1
&?? 2
= ?? 2
+ ?? ?? 2
 are equal if and only if their real and 
imaginary parts are equal respectively 
i.e. ??? 1
= ?? 2
 
? ????? ?(?? 1
) = ???? ?(?? 2
) and ?? ?? (?? 1
) = ?? ?? (?? 2
). 
Problem 2 : Find the value of ?? and ?? for which 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3- 5?? )+ (1+ 2???? ), where ?? ,???? 
Solution : 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3 - 5?? )+ (1 + 2???? ) 
???? 4
- 3?? 2
- 4 = 0 ? ?? 2
= 4 ? ?? = ±2 
and ?2?? - ?? = -5+ 2?? 
2?? + 5 = 3?? 
when ?? = 2 ? ?? = 3 
and ?? = -2 ? ??? = 1/3? Ans. (2,3) or (-2,1/3) 
Problem 3 : Find the value of expression ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3, when ?? = -1 + ?? . 
Solution : ??? = -1 + ?? 
(?? + 1)
2
= ?? 2
 
?? 2
+ 2?? + 2 = 0 
now, ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3 = (?? 2
+ 2?? + 2)(?? 2
+ 2?? - 1)+ 5 = 5 
Problem 4 : Find the square root of -21- 20 
Solution : ? Let ?? + ???? = v-21- 20?? 
??(?? + ???? )
2
= -21- 20?? ???? 2
- ?? 2
= -21?????? = -10?? ???????? ?(?? )?\&?(???? ) ???? 2
= 4 ? ??? = ±2?? ?? h???? ?? = 2,?? = -5 ?????? ?? = -2,?? = 5???? + ????
= (2- ?? 5) ???? (-2+ ?? 5)? 
Representation of a complex number : 
To each complex number there corresponds one and only one point in plane, and 
conversely to each point in the plane there corresponds one and only one complex 
number. Because of this we often refer to the complex number ?? as the point ?? . 
(a) Cartesian Form (Geometric Representation): 
Every complex number ?? = ?? + ?? y can be represented by a point on the Cartesian plane 
known as complex plane (Argand diagram) by the ordered pair ( ?? ,?? ). 
 
Length OP is called modulus of the complex number which is denoted by |?? |&?? is called 
argument or amplitude. 
|?? | = v?? 2
+ ?? 2
 and ?????? ??? = (
?? ?? ) (angle made by OP with positive ?? -axis) 
Note : 
(i) Argument of a complex number is a many valued function. If ?? is the argument of 
a complex number then 2???? + ?? ;?? ? ?? will also be the argument of that complex number. 
Any two arguments of a complex number differ by 2???? . 
(ii) The unique value of ?? such that -?? < ?? = ?? is called the principal value of the 
argument. 
Unless otherwise stated, amp z implies principal value of the argument. 
(iii) By specifying the modulus & argument a complex number is defined completely. For 
the complex number 0+ 0 i the argument is not defined and this is the only complex 
number which is only given by its modulus. 
(b) Trignometric/Polar Representation : 
?? = ?? (?????? ??? + ???????? ??? ) where |?? | = ?? ;?????? ??? = ?? ;?? ? = ?? (?????? ??? - ???????? ??? ) 
Note : ?????? ??? + ???????? ??? is also written as ?????? ??? 
(c) Euler's Formula : 
?? = ?? ?? ????
,|?? | = ?? ,?????? ??? = ?? 
?? ? = ?? ?? -????
 
?? ????
= ?????? ??? + ???????? ??? 
Note : If ?? is real then ?????? ??? =
?? ????
+?? -????
2
;?????? ??? =
?? ????
-?? -????
2?? 
Page 5


Complex Numbers 
Complex numbers, an extension of the real numbers, are fundamental in various branches of 
mathematics, science, and engineering. Introduced to solve equations that do not have real 
solutions, complex numbers have since become a critical tool in numerous theoretical and 
practical applications. They originated from the need to find solutions to quadratic equations 
where the discriminant (the term under the square root) was negative, thereby indicating no 
real roots. This necessity led mathematicians to define the imaginary unit iii, where i2=-1i^2 = -
1i2=-1, allowing for the creation of numbers of the form a+bia + bia+bi, where a and b are real 
numbers. Complex numbers are now indispensable in fields such as electrical engineering, 
quantum mechanics, and signal processing, demonstrating their profound utility beyond their 
theoretical origins. 
The complex number system 
A complex number (?? ) is a number that can be expressed in the form ?? = ?? + ???? where ?? 
and ?? are real numbers and ?? 2
= -1. Here  'a' is called as real part of ?? which is denoted 
by (???? ??? ) and ' ?? ' is called as imaginary part of ?? , which is denoted by (Im z). 
Any complex number is : 
(i) Purely real, if ?? = 0 
(ii) Imaginary, if ?? ? 0. 
(iii) Purely imaginary, if ?? = 0 
Note : 
(a) The set ?? of real numbers is a proper subset of the Complex Numbers. Hence the 
complete number system is ?? ? ?? ? ?? ? ?? ? ?? ? ?? . 
(b) Zero is purely real as well as purely imaginary but not imaginary. 
(c) ?? = v-1 is called the imaginary unit. 
 ???????? ?? 2
= -1;?? 3
= -?? ;?? 4
= 1 ?????? .  
(d) v?? v?? = v???? only if atleast one of ?? or ?? is non-negative. 
(e) If ?? = ?? + ???? , then ?? - ???? is called complex conjugate of ?? and written as ?? ? = ?? - ???? 
(f) Real numbers satisfy order relations, where as imaginary numbers do not satisfy 
order relations i.e. ??? > 0,3+ ?? < 2 are meaningless. 
Fundamental operations with complex 
numbers 
In performing operations with complex numbers we can proceed as in the algebra of real 
numbers, replacing ?? 2
 by -1 when it occurs. 
1. Addition (?? + ???? )+ (?? + ???? ) = ?? + ???? + ?? + ???? = (?? + ?? )+ (?? + ?? )?? 
2. Subtraction (?? + ???? )- (?? + ???? )= ?? + ???? - ?? - ???? = (?? - ?? )+ (?? - ?? )?? 
3. Multiplication (?? + ???? )(?? + ???? )= ???? + ?????? + ?????? + ???? ?? 2
= (???? - ???? )+ (???? + ???? )?? 
4. Division 
?? +????
?? +????
=
?? +????
?? +????
·
?? -????
?? -????
=
???? -?????? +?????? -???? ?? 2
?? 2
-?? 2
?? 2
=
???? +???? +(???? -???? )?? ?? 2
+?? 2
=
???? +????
?? 2
+?? 2
+
???? -????
?? 2
+?? 2
?? 
Inequalities in imaginary numbers are not defined. There is no validity if we say that 
imaginary number is positive or negative. 
e.g. ??? > 0,4+ 2?? < 2 + 4 i are meaningless. 
In real numbers if ?? 2
+ ?? 2
= 0 then ?? = 0 = ?? however in complex numbers, 
?? 1
2
+ ?? 2
2
= 0 ???????? ??????? ??????????? ?? 1
= ?? 2
= 0 
Problem 1 : Find the multiplicative inverse of 4+ 3?? . 
Solution : Let ?? be the multiplicative inverse of 4 + 3?? then 
?? (4 + 3?? )= 1 
?? =
1
4+ 3?? ×
4 - 3?? 4 - 3?? =
4 - 3?? 16+ 9
=
4- 3?? 25
? ?????? . ?
4 - 3?? 25
 
Equality In Complex Number : 
Two complex numbers ?? 1
= ?? 1
+ ?? ?? 1
&?? 2
= ?? 2
+ ?? ?? 2
 are equal if and only if their real and 
imaginary parts are equal respectively 
i.e. ??? 1
= ?? 2
 
? ????? ?(?? 1
) = ???? ?(?? 2
) and ?? ?? (?? 1
) = ?? ?? (?? 2
). 
Problem 2 : Find the value of ?? and ?? for which 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3- 5?? )+ (1+ 2???? ), where ?? ,???? 
Solution : 
(?? 4
+ 2???? )- (3?? 2
+ ???? ) = (3 - 5?? )+ (1 + 2???? ) 
???? 4
- 3?? 2
- 4 = 0 ? ?? 2
= 4 ? ?? = ±2 
and ?2?? - ?? = -5+ 2?? 
2?? + 5 = 3?? 
when ?? = 2 ? ?? = 3 
and ?? = -2 ? ??? = 1/3? Ans. (2,3) or (-2,1/3) 
Problem 3 : Find the value of expression ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3, when ?? = -1 + ?? . 
Solution : ??? = -1 + ?? 
(?? + 1)
2
= ?? 2
 
?? 2
+ 2?? + 2 = 0 
now, ?? 4
+ 4?? 3
+ 5?? 2
+ 2?? + 3 = (?? 2
+ 2?? + 2)(?? 2
+ 2?? - 1)+ 5 = 5 
Problem 4 : Find the square root of -21- 20 
Solution : ? Let ?? + ???? = v-21- 20?? 
??(?? + ???? )
2
= -21- 20?? ???? 2
- ?? 2
= -21?????? = -10?? ???????? ?(?? )?\&?(???? ) ???? 2
= 4 ? ??? = ±2?? ?? h???? ?? = 2,?? = -5 ?????? ?? = -2,?? = 5???? + ????
= (2- ?? 5) ???? (-2+ ?? 5)? 
Representation of a complex number : 
To each complex number there corresponds one and only one point in plane, and 
conversely to each point in the plane there corresponds one and only one complex 
number. Because of this we often refer to the complex number ?? as the point ?? . 
(a) Cartesian Form (Geometric Representation): 
Every complex number ?? = ?? + ?? y can be represented by a point on the Cartesian plane 
known as complex plane (Argand diagram) by the ordered pair ( ?? ,?? ). 
 
Length OP is called modulus of the complex number which is denoted by |?? |&?? is called 
argument or amplitude. 
|?? | = v?? 2
+ ?? 2
 and ?????? ??? = (
?? ?? ) (angle made by OP with positive ?? -axis) 
Note : 
(i) Argument of a complex number is a many valued function. If ?? is the argument of 
a complex number then 2???? + ?? ;?? ? ?? will also be the argument of that complex number. 
Any two arguments of a complex number differ by 2???? . 
(ii) The unique value of ?? such that -?? < ?? = ?? is called the principal value of the 
argument. 
Unless otherwise stated, amp z implies principal value of the argument. 
(iii) By specifying the modulus & argument a complex number is defined completely. For 
the complex number 0+ 0 i the argument is not defined and this is the only complex 
number which is only given by its modulus. 
(b) Trignometric/Polar Representation : 
?? = ?? (?????? ??? + ???????? ??? ) where |?? | = ?? ;?????? ??? = ?? ;?? ? = ?? (?????? ??? - ???????? ??? ) 
Note : ?????? ??? + ???????? ??? is also written as ?????? ??? 
(c) Euler's Formula : 
?? = ?? ?? ????
,|?? | = ?? ,?????? ??? = ?? 
?? ? = ?? ?? -????
 
?? ????
= ?????? ??? + ???????? ??? 
Note : If ?? is real then ?????? ??? =
?? ????
+?? -????
2
;?????? ??? =
?? ????
-?? -????
2?? 
(d) Vectorial Representation : 
Every complex number can be considered as the position vector of a point. If the point ?? 
represents the complex number ?? then, ????
????? 
= ?? & |????
????? 
| = |?? |. 
Argument of a Complex Number : 
Argument of a non-zero complex number ?? (?? ) is denoted and defined by ?????? ?(?? )= angle 
which OP makes with the positive direction of real axis. 
If ???? = |?? | = ?? and ?????? ?(?? ) = ?? , then obviously ?? = ?? (?????? ??? + ???????? ??? ), called the polar 
form of ?? . 
'Argument of ?? ' would mean principal argument of ?? (i.e. argument lying in (-?? ,?? ] 
unless the context requires otherwise. Thus argument of a complex number ?? = ?? + ???? =
?? (?????? ??? + ???????? ??? ) is the value of ?? satisfying ???????? ??? = ?? and ???????? ??? = ?? . Let ?? = ?????? -1
?|
?? ?? | 
(i) ??? > 0,?? > 0 
 
p.v. ?????? ??? = ?? 
(ii) ?? = 0,?? > 0