Properties of complex numbers

1. Commutative property:

Addition of two complex numbers is commutative.

If z represents a complex numbers then z₁ = (a,b) and z₂ = (c,d) are two complex numbers. For commutative property of addition

z₁+z₂ = z₂ + z

Now we will prove the commutative property.

z₁ = (a,b) = a+ib   and   z₂ = (c,d) = c+id

z₁ + z₂  = (a+ib) + (c+id)

=  a+c +i(b+d) [adding real and imaginary parts]

=  c+a +i(d+b)

=  z₂ + z

2. Associative property:

Addition of three complex numbers is associative.

If z₁ = (a,b), z₂ = (c,d) and z₃ =(e,f) then

z₁+( z₂+z₃)  = ( z₁+ z₂)+ z

Now we will prove the associative property.

z₁+( z₂+z₃)  =   (a+ib)+[(c+id)+(e+if)]

=   (a+ib)+[(c+e)+i(d+f)]

=   (a+c+e)+i(b+d+f)

=   [(a+c)+i(b+d)]+(e+if)

=   [(a+ib)+(c+id)]+(e+if)

=    ( z₁+ z₂)+ z

3. Existence of zero (additive identity):

The complex number (0,0) is the additive identity for addition of complex numbers. (0,0) is called as zero complex number.

If z= (a,b) and 0= (0,0), then

z+0 = 0+z = z

4.Existence of inverse:

The negative complex number is the additive inverse of a complex number.

If z= (a,b) then the inverse of this complex number is

-z= (-a,-b).

z+(-z)  =  (a,b)+(-a,-b)

=   (a-a, b-b)

=   0 =  (-z)+z

Thus z+(-z) = 0 = (-z)+z

Properties of multiplication

We had seen properties of addition, now we are going to see properties of complex numbers for multiplication.

1.Commutative property:

Multiplication of two complex numbers is commutative.

If z represents a complex numbers then z₁ = (a,b) and z₂ = (c,d) are two complex numbers. For commutative property of multiplication

z₁ . z₂ = z₂ . z

Now we will prove the commutative property.

z₁ = (a,b) = a+ib   and   z₂ = (c,d) = c+id

z₁ . z₂  = (a+ib) . (c+id)

By the definition of multiplication of complex numbers

=  [(ca-db)+i(da+cb)] (since multiplication of two numbers is commutative, we can change the order of multiplication.)

= z₂ . z₁.

2.Associative property:

Multiplication of three complex numbers is associative.

If z₁ = (a,b), z₂ = (c,d) and z₃ =(e,f) then

z₁.( z₂.z₃)  = ( z₁. z₂). z

Now we will prove the associative property.

z₁.( z₂.z₃)  =   (a+ib).[(c+id).(e+if)]

=   (a+ib).[(ce-df)+i(cf+de)]

=   [a(ce-df)-b(cf+de)]+i[a(cf+de)+b(ce-df)]

( z₁. z₂). z₃ =   [(a+ib).(c+id)].(e+if)

I and II are same. So z₁.( z₂.z₃)  = ( z₁. z₂). z

3. Existence of identity element for multiplication:

The complex number (1,0) is the identity or unity for multiplication.

If z= (a,b) then z.(1,0) = (a+ib).(1+i0)

=  (a.1-b.0)+i(a.0+b.1)

=  (a+ib)

=    z = (1,0).z

4. Existence of inverse:

If z = a+ib ≠ (0,0), then it has the multiplicative inverse. Now let us find the inverse for a+ib

Let c+id be the complex number such that

(a+ib).(c+id) = (1+i0)

Equating the real and imaginary parts

ac-bd = 1 and ad+bc = 0

Solving for c and d, we get

c = a/(a²+b²)       and d = -b/(a²+b²)

so inverse of z = a/(a²+b²)-ib/(a²+b²)

5.Distributive laws:

If z₁,z₂,z₃ are any three complex numbers which are not equal to zero, then

z₁.(z₂+z₃)  =  z₁.z₂ +z₁.z₃

Let z₁ = a+ib, z₂ = c+id and z₃= e+if, then

z₁.(z₂+z₃)      =   a+ib.[(c+id)+(e+if)]

=   a+ib.(c+e)+i(d+f)]

=   a.(c+e)-b.(d+f)+i[a.(d+f)+b(c+e)]