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IMAGINARY NUMBERS 
Square root of a negative number is called imaginary number. While solving the equations ?? 2
+1=0, a 
quantity v-1 is obtained and is denoted by ?? (iota) which is imaginary. Further v-2 is an imaginary 
number and can be written as 
v-2=v2×v-1=v2?? 
If ?? <0, then v?? =v|?? |?? 
  
Page 2


 
 
 
 
IMAGINARY NUMBERS 
Square root of a negative number is called imaginary number. While solving the equations ?? 2
+1=0, a 
quantity v-1 is obtained and is denoted by ?? (iota) which is imaginary. Further v-2 is an imaginary 
number and can be written as 
v-2=v2×v-1=v2?? 
If ?? <0, then v?? =v|?? |?? 
  
 
 
 
 
 
Integral Powers of ?? 
We have ?? =v-1 so ?? 2
=-1,?? 3
=-?? ,?? 4
=1 
For any ?? ??? , we have 
?? 4?? +1
=?? ,?? 4?? +2
=-1
?? 4?? +3
=-?? ,?? 4?? =1
 
Thus any integral power of ?? can be expressed as ±1 or ±?? . 
In other words ?? ?? ={
(-1)
?? /2
, if ?? is even integer 
(-1)
?? -1
2
?? , if ?? is odd integer 
. 
Also ?? -?? =
1
?? ?? 
  
Page 3


 
 
 
 
IMAGINARY NUMBERS 
Square root of a negative number is called imaginary number. While solving the equations ?? 2
+1=0, a 
quantity v-1 is obtained and is denoted by ?? (iota) which is imaginary. Further v-2 is an imaginary 
number and can be written as 
v-2=v2×v-1=v2?? 
If ?? <0, then v?? =v|?? |?? 
  
 
 
 
 
 
Integral Powers of ?? 
We have ?? =v-1 so ?? 2
=-1,?? 3
=-?? ,?? 4
=1 
For any ?? ??? , we have 
?? 4?? +1
=?? ,?? 4?? +2
=-1
?? 4?? +3
=-?? ,?? 4?? =1
 
Thus any integral power of ?? can be expressed as ±1 or ±?? . 
In other words ?? ?? ={
(-1)
?? /2
, if ?? is even integer 
(-1)
?? -1
2
?? , if ?? is odd integer 
. 
Also ?? -?? =
1
?? ?? 
  
 
 
 
Note: 
v?? ×v?? =v???? is true if at least one of ?? and ?? are non-negative. If ?? <0 and ?? <0, then 
v?? ×v?? =v-|?? |×v-|?? |
 =?? v|?? |×?? v|?? |
 =-v????
 
Thus v???? =v?? ·v?? only when at most one of ?? & ?? is negative 
  
Page 4


 
 
 
 
IMAGINARY NUMBERS 
Square root of a negative number is called imaginary number. While solving the equations ?? 2
+1=0, a 
quantity v-1 is obtained and is denoted by ?? (iota) which is imaginary. Further v-2 is an imaginary 
number and can be written as 
v-2=v2×v-1=v2?? 
If ?? <0, then v?? =v|?? |?? 
  
 
 
 
 
 
Integral Powers of ?? 
We have ?? =v-1 so ?? 2
=-1,?? 3
=-?? ,?? 4
=1 
For any ?? ??? , we have 
?? 4?? +1
=?? ,?? 4?? +2
=-1
?? 4?? +3
=-?? ,?? 4?? =1
 
Thus any integral power of ?? can be expressed as ±1 or ±?? . 
In other words ?? ?? ={
(-1)
?? /2
, if ?? is even integer 
(-1)
?? -1
2
?? , if ?? is odd integer 
. 
Also ?? -?? =
1
?? ?? 
  
 
 
 
Note: 
v?? ×v?? =v???? is true if at least one of ?? and ?? are non-negative. If ?? <0 and ?? <0, then 
v?? ×v?? =v-|?? |×v-|?? |
 =?? v|?? |×?? v|?? |
 =-v????
 
Thus v???? =v?? ·v?? only when at most one of ?? & ?? is negative 
  
 
 
 
 
COMPLEX NUMBERS 
The formal addition, ' ?? +???? ', where ?? ,?? ??? and the collection of all such expressions is called the set of 
complex numbers. For the complex number, ?? =?? +???? ,
'
 ?? '
 is called as real part of ?? and is denoted by 
Re (?? ) while ' ?? ' is called as imaginary part of ?? and is denoted by Im (?? ) . 
Set of complex numbers ?? is said to be purely real if ?? ?? (?? )=0 and is said to be purely imaginary if 
Re (?? )=0. The complex number 0+0?? =0 is both purely real and purely imaginary. 
All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or may 
not be purely imaginary. 
For e.g. 4+3?? is imaginary but not purely imaginary. 
  
Page 5


 
 
 
 
IMAGINARY NUMBERS 
Square root of a negative number is called imaginary number. While solving the equations ?? 2
+1=0, a 
quantity v-1 is obtained and is denoted by ?? (iota) which is imaginary. Further v-2 is an imaginary 
number and can be written as 
v-2=v2×v-1=v2?? 
If ?? <0, then v?? =v|?? |?? 
  
 
 
 
 
 
Integral Powers of ?? 
We have ?? =v-1 so ?? 2
=-1,?? 3
=-?? ,?? 4
=1 
For any ?? ??? , we have 
?? 4?? +1
=?? ,?? 4?? +2
=-1
?? 4?? +3
=-?? ,?? 4?? =1
 
Thus any integral power of ?? can be expressed as ±1 or ±?? . 
In other words ?? ?? ={
(-1)
?? /2
, if ?? is even integer 
(-1)
?? -1
2
?? , if ?? is odd integer 
. 
Also ?? -?? =
1
?? ?? 
  
 
 
 
Note: 
v?? ×v?? =v???? is true if at least one of ?? and ?? are non-negative. If ?? <0 and ?? <0, then 
v?? ×v?? =v-|?? |×v-|?? |
 =?? v|?? |×?? v|?? |
 =-v????
 
Thus v???? =v?? ·v?? only when at most one of ?? & ?? is negative 
  
 
 
 
 
COMPLEX NUMBERS 
The formal addition, ' ?? +???? ', where ?? ,?? ??? and the collection of all such expressions is called the set of 
complex numbers. For the complex number, ?? =?? +???? ,
'
 ?? '
 is called as real part of ?? and is denoted by 
Re (?? ) while ' ?? ' is called as imaginary part of ?? and is denoted by Im (?? ) . 
Set of complex numbers ?? is said to be purely real if ?? ?? (?? )=0 and is said to be purely imaginary if 
Re (?? )=0. The complex number 0+0?? =0 is both purely real and purely imaginary. 
All purely imaginary numbers except zero are imaginary numbers but an imaginary number may or may 
not be purely imaginary. 
For e.g. 4+3?? is imaginary but not purely imaginary. 
  
 
 
 
 
 
Equality of Complex numbers 
Two complex numbers ?? +???? and ?? +???? are said to be equal, if and only if, ?? =?? and ?? =?? . i.e. the 
corresponding real and imaginary parts are equal. 
If ?? +???? =?? 1
 and ?? +???? =?? 2
 
then either ?? 1
=?? 2
 
or ?? 1
??? 2
 
Note: For imaginary numbers, the property of order is not defined because ?? is neither positive, zero nor 
negative. i.e., ?? 1
>?? 2
 or ?? 1
<?? 2
 is meaningless if ?? and ?? are not equal to zero. 
  
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FAQs on Flashcards: Complex Number and Quadratic Equations - Mathematics (Maths) for JEE Main & Advanced

1. How are complex numbers defined and represented in mathematics?
Ans. Complex numbers are represented 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.
2. How do you add and subtract complex numbers?
Ans. To add or subtract complex numbers, simply add or subtract the real parts and the imaginary parts separately.
3. What is the geometric interpretation of a complex number?
Ans. A complex number can be represented as a point in the complex plane, where the real part corresponds to the x-coordinate and the imaginary part corresponds to the y-coordinate.
4. How do you multiply and divide complex numbers?
Ans. To multiply complex numbers, use the distributive property and the fact that i^2 = -1. To divide complex numbers, multiply by the conjugate of the denominator.
5. How are complex numbers used in solving quadratic equations?
Ans. Complex numbers are used to find the roots of quadratic equations, especially when the discriminant is negative. The complex roots come in conjugate pairs due to the nature of the quadratic formula.
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