Flashcards: Complex Number and Quadratic Equations | Flashcards for JEE PDF Download

 Page 1


 
 
 
 
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.