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, COMPLEX NUMBERS
COMPLEX NUMBERS
If ‘a’, ‘b’ are two real numbers, then a number of the form a + ib is called a complex number
Set of complex Numbers : The set of all complex numbers is denoted by C.
i.e. C = {a + ib | a,b R }
Equality of Complex Numbers : Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and
b1 = b2 i.e. Re (z1) = Re(z2) and Im (z1) = Im (z2)
FUNDAMENTAL OPERATIONS ON COMPLEX NUMBERS
ADDITION : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then their sum z1 + z2 is defined as
the complex number (a1 + a2) + i (b1 + b2)
Properties of addition of complex numbers
(i) Addition is commutative : For any two complex numbers z1 and z2, we have
z1 z 2 z 2 z1
(ii) Addition is associative : For any three complex numbers z1, z2, z3 we have
(z1 + z2) + z3 = z1 + (z2 + z3)
(iii) Existence of additive identity : The complex number 0 = 0 + i0 is the identity element for addition i.e.
z + 0 = z = 0 + z for all z C
(iv) Existence of additive inverse : For every complex number z there exists –z such that
z + (–z) = 0 = (–z) + z
The complex number –z is called the additive inverse of z.
Substraction : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then the subtraction of z2 from z1 is
denoted by z1 – z2 and is defined as the addition of z1 and –z2.
Thus, z1 – z2
= (a1 – a2) + i (b1 – b2)
Multiplication : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then, the multiplication of z1 with
z2 is denoted by z1z2 and is defined as the complex number.
(a1a2 – b1 b2) + i (a1b2 + a2b1)
Properties of Multiplication :
(i) Multiplication is commutative. For any two complex numbers z1 and z2, we have
z1 z2 = z2 z1
(ii) Multiplication is associative : For any three complex numbers z1, z2, z3 we have
(z1 z2) z3 = z1 (z2 z3)
(iii) Existence of identity element for multiplication. The complex number 1 = 1 + i0 is the identity element for
multiplication i.e. for every complex number z, we have
z.1=z
(iv) Exitence of multiplicative inverse : Corresponding to every non-zero complex number z = a + ib there exists
a complex number z1 = x + iy such that
1
z . z 1 = 1 z1
z
[1]
JEE Main JEE Adv. BITSAT WBJEE MHT CET and more...
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Rating on Google Play Students using daily Questions available
With MARKS app you can do all these things for free
Solve Chapter-wise PYQ of JEE Main, JEE Advanced, NEET, BITSAT, WBJEE, MHT CET & more
Create Unlimited Custom Tests for any exam
Attempt Top Questions for JEE Main which can boost your rank
Track your exam preparation with Preparation Trackers
Complete daily goals, rank up on the leaderboard & compete with other aspirants
4.8 50,000+ 2,00,000+
Rating on Google Play Students using daily Questions available
, COMPLEX NUMBERS
COMPLEX NUMBERS
If ‘a’, ‘b’ are two real numbers, then a number of the form a + ib is called a complex number
Set of complex Numbers : The set of all complex numbers is denoted by C.
i.e. C = {a + ib | a,b R }
Equality of Complex Numbers : Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and
b1 = b2 i.e. Re (z1) = Re(z2) and Im (z1) = Im (z2)
FUNDAMENTAL OPERATIONS ON COMPLEX NUMBERS
ADDITION : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then their sum z1 + z2 is defined as
the complex number (a1 + a2) + i (b1 + b2)
Properties of addition of complex numbers
(i) Addition is commutative : For any two complex numbers z1 and z2, we have
z1 z 2 z 2 z1
(ii) Addition is associative : For any three complex numbers z1, z2, z3 we have
(z1 + z2) + z3 = z1 + (z2 + z3)
(iii) Existence of additive identity : The complex number 0 = 0 + i0 is the identity element for addition i.e.
z + 0 = z = 0 + z for all z C
(iv) Existence of additive inverse : For every complex number z there exists –z such that
z + (–z) = 0 = (–z) + z
The complex number –z is called the additive inverse of z.
Substraction : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then the subtraction of z2 from z1 is
denoted by z1 – z2 and is defined as the addition of z1 and –z2.
Thus, z1 – z2
= (a1 – a2) + i (b1 – b2)
Multiplication : Let z1 = a1 + ib1 and z2 = a2 + ib2 be two complex numbers. Then, the multiplication of z1 with
z2 is denoted by z1z2 and is defined as the complex number.
(a1a2 – b1 b2) + i (a1b2 + a2b1)
Properties of Multiplication :
(i) Multiplication is commutative. For any two complex numbers z1 and z2, we have
z1 z2 = z2 z1
(ii) Multiplication is associative : For any three complex numbers z1, z2, z3 we have
(z1 z2) z3 = z1 (z2 z3)
(iii) Existence of identity element for multiplication. The complex number 1 = 1 + i0 is the identity element for
multiplication i.e. for every complex number z, we have
z.1=z
(iv) Exitence of multiplicative inverse : Corresponding to every non-zero complex number z = a + ib there exists
a complex number z1 = x + iy such that
1
z . z 1 = 1 z1
z
[1]
