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.
Read More