Class 12 Exam > Class 12 Videos > Definition, Algebra of complex number (Part -1) - Complex Numbers and Quadratic Equations, Math
Definition, Algebra of complex number (Part -1) - Complex Numbers and Quadratic Equations, Math Video Lecture - Class 12
FAQs on Definition, Algebra of complex number (Part -1) - Complex Numbers and Quadratic Equations, Math Video Lecture - Class 12
| 1. What is a complex number? |  |
Ans. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is an imaginary unit. The real part of the complex number is represented by 'a' and the imaginary part is represented by 'b'. For example, 3 + 4i is a complex number, where 3 is the real part and 4i is the imaginary part.
| 2. How do you add and subtract complex numbers? | |
Ans. To add or subtract complex numbers, we simply add or subtract the real parts and the imaginary parts separately. For example, to add (3 + 4i) and (2 + 5i), we add 3 + 2 and 4i + 5i, which gives us 5 + 9i. Similarly, to subtract (3 + 4i) from (2 + 5i), we subtract 3 from 2 and 4i from 5i, which gives us -1 + i.
| 3. What is the algebraic form of a complex number? | |
Ans. The algebraic form of a complex number is the expression a + bi, where a and b are real numbers, and i is the imaginary unit. In this form, the real part is represented by 'a' and the imaginary part is represented by 'b'. For example, the algebraic form of the complex number 3 + 4i is 3 + 4i.
| 4. How do you multiply complex numbers? | |
Ans. To multiply complex numbers, we use the distributive property and the fact that i^2 = -1. Let's say we want to multiply (a + bi) and (c + di). We can expand this using the distributive property and simplify it using the fact that i^2 = -1. The result will be (ac - bd) + (ad + bc)i. For example, to multiply (3 + 4i) and (2 + 5i), we have (3 * 2 - 4 * 5) + (3 * 5 + 4 * 2)i, which gives us -7 + 22i.
| 5. How do you divide complex numbers? | |
Ans. To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Let's say we want to divide (a + bi) by (c + di). We multiply the numerator and denominator by the conjugate of (c + di), which is (c - di). After simplifying, the result will be [(ac + bd) + (bc - ad)i] / (c^2 + d^2). For example, to divide (3 + 4i) by (2 + 5i), we multiply the numerator and denominator by the conjugate of (2 + 5i), which is (2 - 5i). After simplifying, we get (2 + 23i) / 29.
Text Transcript from Video
hello friends this video on complex
number part 1 is brought to you by exam
clear calm no more fear from exam in
this lesson we'll study complex number
and quality equations definition of
complex numbers algebra of complex
numbers is an addition difference
multiplication and division identities
modulus and conjugate of complex numbers
Argan plane and polar representation of
complex number quadratic equations let
us define complex number complex number
is any number in the form of a plus IB
where is real and B is imaginary part
very complex number is any number in the
form of e plus I B where a is the real
part and B is the imaginary part and
what is I I is root of -1 is root of -1
complex number is any number in the form
of a plus IV where a is real Part B is
measuring part and ie root of -1 for
example 3 plus 2 plus 3 is 7 + i9 etc so
what is the use of complex numbers
complex numbers are used in scientific
fields also if we have two complex
number and if they are equal then the
real parts are equal and imaginary parts
are also equal if you want to represent
the complex number in the XY coordinate
you can represent like it's a in x-axis
a be in my X's and you get a plus B I so
when I'm saying two complex number are
equal only real parts are equal and
imagery parts like for example I say z1
is equal to a plus bi and z2 is equal to
red suppose C plus D I I am when I am
saying that z1 is equal to z2 this
implies that e is equal to C and B is
equal to C now let's discuss algebra of
complex number two complex number can be
added they can be subtracted we can
multiply it to complex number and we can
divide two complex and it will discuss
all this that means we can add divide
multiply and subtract
to complex um let's take example of
addition of two complex on way that
suppose they are two complex numbers
Adrian's been to a plus VA n 0 then Z 1
plus Z 2 is equal to a plus C less I be
lusty example if we have 7 is equal to 2
plus I 3 and Z 2 is going to homeless
I fine we have to add this then we will
say Z 1 plus I 2 is equal to 2 plus 4 2
plus 4 plus I into 3 plus 5 3 + 5 and
this is equal to 6 plus I so what we
have done we have added the real part
and then we have added up imaginary part
now the addition of complex numbers
should satisfy the following properties
the keys of Troezen log Z 1 plus Z 2 is
also a complex number if you add two
complex number you say Z 1 is a complex
number Z 2 is a complex number then you
get Deb 3 this is also complex number
and this is a to the property you add
two complex than one what if it is a
complex number
community along you say as I'd 1 plus Z
2 or Z 2 plus Z what will give you same
result then we have associative law when
we add veteran Z to first and then add Z
3 all we add Z 1 that led to her three
first and then adds Edwin you get them
same as L this is called associative law
then we have identity there is a number
0 when you add that number to complex
the world you get the same complex
number C ready you have additive inverse
that means for every complex number you
have a number minus Z that is if you add
that number to the symbol becomes 0 so
for every number you have a negative of
a number for every complex number you
have a negative a complex number and if
you add that number to the negative of
the number you get 0 so please
understand we have these five properties
defined for addition close allow you add
two complex symbol you
example commutative law you had you see
a Zen to present to our jet to Brazil
for the same associative law editor
parenting and native inverse now let's
see difference of two complex number if
they are two complex numbers Z 1 and Z 2
then the difference of Z 1 Z 2 is if 0
to a plus IB and Z 2 is equal to C plus
I D then the difference is you subtract
the real parts and then you subtract an
imaginary part for example we have 6
plus 3i + 2 + I and do you want to
subtract this so what we get is 6 minus
2 plus I into 3 minus 1 and this is
going to 4 plus 2i Russia is similar to
addition here instead of doing plus we
are doing - let's see multiplication and
they've been to complex number Z 1 and Z
2 then we say Zettler into Z 2 is equal
to AC minus VT less I into a deeper
specie that looks complex but actually
it is silly
let's take an example so we have 3 + i5
and we want to multiply this with 2 + i6
what we'll do we'll take the light
dispersed C plus I Phi into 2 plus is
normal simple multiplication will take
first number of this multiplied with
this 3 into 2 is 6 plus will take we
multiply this 3 into PI 6 YK / ticket I
into 6 into 3 then it takes second
number of this that is I into Phi will
multiply with this 2 and then we say I
into 5 multiplied by I into 6 so what we
get is 6 plus 80 9 + 10 I minus 30y this
is minus Fe because 6 e
by 30 and I is equal to root out minus 1
as we have seen so I square is going to
minus 1
so I into y in comes minus 1 and 60 - 5
becomes 30 so it is 60 minus 30 plus I
into 18 plus 10 so this is exactly what
is written here this is AC this is minus
BD this is AD and this is VC you do not
learn this you can just simply do a
multiplication and you'll get there is
minus 24 plus I into D and this is the
answer
so multiplication of complex numbers
satisfy the following properties first
is closer lock you multiply two complex
number what you get is a complex number
you multiply you say 7% or Z 2 into 7 it
is all on the same you first multiply
two complex number and then
multi-layered complex number or you
first multiply 2nd and 3rd complex
number then multiply first complex
number too late you get the same result
similarly for every complex number you
have a irritative identity is 1 you
multiply the complex number with 1 you
get seen from this symbol there is
something called multiplicative inverse
this this is important for every complex
number said there is a complex number
called 1 by Z and you multiply 1 by Z to
Z you get 1 and please make sure Z
should not be equal to 0 because if it
said is going to 0 1 by Z is infinite
also distribute all holds 2 for
multiplication you say Z 1 into Z 2 plus
Z 3 or Z 1 Z 2 plus Z 3 both are same
let us take a problem where we are
supposed to find our multiplicative
inverse so we have to fess up we are
supposed to find a multiplicative
inverse of 1 2 minus 3i so if you see as
Z is equal to 2 minus 3i then from the
definition we know multiplicative
integral 1 by Z and that is equal to 1
by 2 minus 3i so let's solve this 1 by 2
minus 3i will rationalize this what is
oscillation you use
and you divide an ad with 2 plus 3i so
what is a summation if you have number
in the form a minus IB so what you do
you add and some you multiply and divide
with a plus IB so what happens a plus I
B into a minus IB if you solve this
becomes a square plus B Square so
whatever you get in numerator is not the
complex number it's a real number and
then you get complex number in a minute
similarly if you have 1 by a plus IB if
you want to rationalize you have to
multiply and divide by E minus IV a
minus F similarly in this case we had 2
minus 3i shimbun percept in this case we
have 2 minus 3i so nationalized by
multiplying and divided by 2 plus CI so
what we get is 2 minus 3i into 2 plus 3i
that becomes 2 square plus 3 square why
it becomes 2 square plus Z square a plus
B a minus B is equal to a square minus B
Square so this should be equal to
actually Q square minus 3i minus 3i
square is equal to 2 square minus 3i
square is nothing but plus 3 square
because I square is minus 1 so that's
why we have got this let's become 2 plus
3i divided by this if you expand this 2
square that is 4 plus 3 square mines bit
of 14 this is about 2 2 by 40 plus 3i by
sorry
34 plus 9 is 30 plus 10 is 30 so this is
the answer light once again 2 by 13 plus
3i by okay this is that you find a
multiplicative miss you write 1 by Z so
this is 1 by Z and 1 by Z you whatever
you get you rationalize so that in
elementary should give only a real
number and is the answer
let's take example of division of
complex number so there are two complex
numbers Z 1 and Z 2 and Z 2 is not equal
to 0 because you don't want to divide by
0 siphon by Z 2 is nothing but Z 1 into
1 positive
and we know how to find one by Z to the
baby upon the last example example we
have Z 1 as well 2 plus V I and J 2 is
going to 2 plus 2 I we have to divide
this so what we get is 2 plus 3i divided
by 2 plus 2i
now we have to rationalize this it is in
the form of a plus IB so we'll multiply
and divide by a minus I P and is 2 minus
2y so this becomes 2 square plus 2
square as we have seen and this we have
to know multiply so we say to me no 2 is
4 just in the business for and then we
say 2 into minus 2y that will become
minus 4i now say take 3i 3i into 2 is 6
I and then we'll say minus 3i into 2 I
minus 3i into to Y so what we get is
this is 4 plus 4 is 8
but here we get is 4 minus 4i plus 6i is
going to 2i this is minus 6i squared so
I squared is equal to minus 1 so this
will become plus 6 so the answer is 10
plus 2i by a or you can say 5 by 4 plus
I by this is the next you have seen what
we have done we have we are at a number
we have got this and now we rationalize
this and then we solve this to get thank
you visit exam clear calm to watch free
educational videos try free online test
it the best quality study material study
from the best tutor and mentors and much
forum times once again
number part 1 is brought to you by exam
clear calm no more fear from exam in
this lesson we'll study complex number
and quality equations definition of
complex numbers algebra of complex
numbers is an addition difference
multiplication and division identities
modulus and conjugate of complex numbers
Argan plane and polar representation of
complex number quadratic equations let
us define complex number complex number
is any number in the form of a plus IB
where is real and B is imaginary part
very complex number is any number in the
form of e plus I B where a is the real
part and B is the imaginary part and
what is I I is root of -1 is root of -1
complex number is any number in the form
of a plus IV where a is real Part B is
measuring part and ie root of -1 for
example 3 plus 2 plus 3 is 7 + i9 etc so
what is the use of complex numbers
complex numbers are used in scientific
fields also if we have two complex
number and if they are equal then the
real parts are equal and imaginary parts
are also equal if you want to represent
the complex number in the XY coordinate
you can represent like it's a in x-axis
a be in my X's and you get a plus B I so
when I'm saying two complex number are
equal only real parts are equal and
imagery parts like for example I say z1
is equal to a plus bi and z2 is equal to
red suppose C plus D I I am when I am
saying that z1 is equal to z2 this
implies that e is equal to C and B is
equal to C now let's discuss algebra of
complex number two complex number can be
added they can be subtracted we can
multiply it to complex number and we can
divide two complex and it will discuss
all this that means we can add divide
multiply and subtract
to complex um let's take example of
addition of two complex on way that
suppose they are two complex numbers
Adrian's been to a plus VA n 0 then Z 1
plus Z 2 is equal to a plus C less I be
lusty example if we have 7 is equal to 2
plus I 3 and Z 2 is going to homeless
I fine we have to add this then we will
say Z 1 plus I 2 is equal to 2 plus 4 2
plus 4 plus I into 3 plus 5 3 + 5 and
this is equal to 6 plus I so what we
have done we have added the real part
and then we have added up imaginary part
now the addition of complex numbers
should satisfy the following properties
the keys of Troezen log Z 1 plus Z 2 is
also a complex number if you add two
complex number you say Z 1 is a complex
number Z 2 is a complex number then you
get Deb 3 this is also complex number
and this is a to the property you add
two complex than one what if it is a
complex number
community along you say as I'd 1 plus Z
2 or Z 2 plus Z what will give you same
result then we have associative law when
we add veteran Z to first and then add Z
3 all we add Z 1 that led to her three
first and then adds Edwin you get them
same as L this is called associative law
then we have identity there is a number
0 when you add that number to complex
the world you get the same complex
number C ready you have additive inverse
that means for every complex number you
have a number minus Z that is if you add
that number to the symbol becomes 0 so
for every number you have a negative of
a number for every complex number you
have a negative a complex number and if
you add that number to the negative of
the number you get 0 so please
understand we have these five properties
defined for addition close allow you add
two complex symbol you
example commutative law you had you see
a Zen to present to our jet to Brazil
for the same associative law editor
parenting and native inverse now let's
see difference of two complex number if
they are two complex numbers Z 1 and Z 2
then the difference of Z 1 Z 2 is if 0
to a plus IB and Z 2 is equal to C plus
I D then the difference is you subtract
the real parts and then you subtract an
imaginary part for example we have 6
plus 3i + 2 + I and do you want to
subtract this so what we get is 6 minus
2 plus I into 3 minus 1 and this is
going to 4 plus 2i Russia is similar to
addition here instead of doing plus we
are doing - let's see multiplication and
they've been to complex number Z 1 and Z
2 then we say Zettler into Z 2 is equal
to AC minus VT less I into a deeper
specie that looks complex but actually
it is silly
let's take an example so we have 3 + i5
and we want to multiply this with 2 + i6
what we'll do we'll take the light
dispersed C plus I Phi into 2 plus is
normal simple multiplication will take
first number of this multiplied with
this 3 into 2 is 6 plus will take we
multiply this 3 into PI 6 YK / ticket I
into 6 into 3 then it takes second
number of this that is I into Phi will
multiply with this 2 and then we say I
into 5 multiplied by I into 6 so what we
get is 6 plus 80 9 + 10 I minus 30y this
is minus Fe because 6 e
by 30 and I is equal to root out minus 1
as we have seen so I square is going to
minus 1
so I into y in comes minus 1 and 60 - 5
becomes 30 so it is 60 minus 30 plus I
into 18 plus 10 so this is exactly what
is written here this is AC this is minus
BD this is AD and this is VC you do not
learn this you can just simply do a
multiplication and you'll get there is
minus 24 plus I into D and this is the
answer
so multiplication of complex numbers
satisfy the following properties first
is closer lock you multiply two complex
number what you get is a complex number
you multiply you say 7% or Z 2 into 7 it
is all on the same you first multiply
two complex number and then
multi-layered complex number or you
first multiply 2nd and 3rd complex
number then multiply first complex
number too late you get the same result
similarly for every complex number you
have a irritative identity is 1 you
multiply the complex number with 1 you
get seen from this symbol there is
something called multiplicative inverse
this this is important for every complex
number said there is a complex number
called 1 by Z and you multiply 1 by Z to
Z you get 1 and please make sure Z
should not be equal to 0 because if it
said is going to 0 1 by Z is infinite
also distribute all holds 2 for
multiplication you say Z 1 into Z 2 plus
Z 3 or Z 1 Z 2 plus Z 3 both are same
let us take a problem where we are
supposed to find our multiplicative
inverse so we have to fess up we are
supposed to find a multiplicative
inverse of 1 2 minus 3i so if you see as
Z is equal to 2 minus 3i then from the
definition we know multiplicative
integral 1 by Z and that is equal to 1
by 2 minus 3i so let's solve this 1 by 2
minus 3i will rationalize this what is
oscillation you use
and you divide an ad with 2 plus 3i so
what is a summation if you have number
in the form a minus IB so what you do
you add and some you multiply and divide
with a plus IB so what happens a plus I
B into a minus IB if you solve this
becomes a square plus B Square so
whatever you get in numerator is not the
complex number it's a real number and
then you get complex number in a minute
similarly if you have 1 by a plus IB if
you want to rationalize you have to
multiply and divide by E minus IV a
minus F similarly in this case we had 2
minus 3i shimbun percept in this case we
have 2 minus 3i so nationalized by
multiplying and divided by 2 plus CI so
what we get is 2 minus 3i into 2 plus 3i
that becomes 2 square plus 3 square why
it becomes 2 square plus Z square a plus
B a minus B is equal to a square minus B
Square so this should be equal to
actually Q square minus 3i minus 3i
square is equal to 2 square minus 3i
square is nothing but plus 3 square
because I square is minus 1 so that's
why we have got this let's become 2 plus
3i divided by this if you expand this 2
square that is 4 plus 3 square mines bit
of 14 this is about 2 2 by 40 plus 3i by
sorry
34 plus 9 is 30 plus 10 is 30 so this is
the answer light once again 2 by 13 plus
3i by okay this is that you find a
multiplicative miss you write 1 by Z so
this is 1 by Z and 1 by Z you whatever
you get you rationalize so that in
elementary should give only a real
number and is the answer
let's take example of division of
complex number so there are two complex
numbers Z 1 and Z 2 and Z 2 is not equal
to 0 because you don't want to divide by
0 siphon by Z 2 is nothing but Z 1 into
1 positive
and we know how to find one by Z to the
baby upon the last example example we
have Z 1 as well 2 plus V I and J 2 is
going to 2 plus 2 I we have to divide
this so what we get is 2 plus 3i divided
by 2 plus 2i
now we have to rationalize this it is in
the form of a plus IB so we'll multiply
and divide by a minus I P and is 2 minus
2y so this becomes 2 square plus 2
square as we have seen and this we have
to know multiply so we say to me no 2 is
4 just in the business for and then we
say 2 into minus 2y that will become
minus 4i now say take 3i 3i into 2 is 6
I and then we'll say minus 3i into 2 I
minus 3i into to Y so what we get is
this is 4 plus 4 is 8
but here we get is 4 minus 4i plus 6i is
going to 2i this is minus 6i squared so
I squared is equal to minus 1 so this
will become plus 6 so the answer is 10
plus 2i by a or you can say 5 by 4 plus
I by this is the next you have seen what
we have done we have we are at a number
we have got this and now we rationalize
this and then we solve this to get thank
you visit exam clear calm to watch free
educational videos try free online test
it the best quality study material study
from the best tutor and mentors and much
forum times once again
About this Video
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Oct 23, 2025
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