Operations on Complex Numbers

The various operations of addition, subtraction, multiplication, division of natural numbers can also be performed for complex numbers also. The details of the various arithmetic operations of complex numbers are as follows.

Addition of Complex Numbers

Th addition of complex numbers is similar to the addition of natural numbers. Here in complex numbers, the real part is added to the real part and the imaginary part is added to the imaginary part. For two complex numbers of the form z1=a+idz1=a+id and z2=c+idz2=c+id, the sum of complex numbers z1+z2=(a+c)+i(b+d)z1+z2=(a+c)+i(b+d).  The complex numbers follow all the following properties of addition.

• Closure Law: The sum of two complex numbers is also a complex number. For two complex numbers z1z1 and z2z2, the sum of z1+z2z1+z2 is also a complex number.
• Commutative Law: For two complex numbers z1z1, z2z2 we have z1+z2=z2+z1z1+z2=z2+z1.
• Associative Law: For the given three complex numbers z1,z2,z3z1,z2,z3 we have z1+(z2+z3)=(z1+z2)+z3z1+(z2+z3)=(z1+z2)+z3.
• Additive Identity: For a complex number z = a + ib, there exists 0 = 0 + i0, such that z + 0 = 0 + z = 0. 
• Additive Inverse: For the complex number z = a + ib, there exists a complex number -z = -a -ib such that z + (-z) = (-z) + z = 0.  Here -z is the additive inverse.

Subtraction of Complex Numbers

The subtraction of complex numbers follows a similar process of subtraction of natural numbers. Here for any two complex numbers, the subtraction is separately performed across the real part and then the subtraction is performed across the imaginary part. For the complex numbers z1z1 = a + ib, z2=c+idz2=c+id, we have z1−z2z1−z2 = (a – c) + i(b – d)

Multiplication of Complex Numbers

The multiplication of complex numbers is slightly different from the multiplication of natural numbers. Here we need to use the formula of i2=−1i2=−1.  For the two complex numbers z1z1 = a + ib, z2z2 = c + id, the product is z1.z2z1.z2 = (ca – bd) + i(ad + bc).   

The multiplication of complex numbers is polar form is slightly different from the above mentioned form of multiplication. Here the absolute values of the two complex numbers are multiplied and their arguments are added to obtain the product of the complex numbers. For the complex numbers z1=r1(Cosθ1+iSinθ1)z1=r1(Cosθ1+iSinθ1), and z₂ = z2=r1(Cosθ2+iSinθ2)z2=r1(Cosθ2+iSinθ2), the product of the complex numbers is z1.z2=r1.r2(Cos(θ1+θ2)+iSin(θ1+θ2))z1.z2=r1.r2(Cos(θ1+θ2)+iSin(θ1+θ2)).

Division of Complex Numbers

The division of complex numbers makes use of the formula of reciprocal of a complex number. For the two complex numbers z1z1 = a + ib, z2z2 = c + id, we have the division as z1z2=(a+ib)×1(c+id)=(a+ib)×(c−id)(c2+d2)z1z2=(a+ib)×1(c+id)=(a+ib)×(c−id)(c2+d2).