Algebraic operation on Complex Numbers - Practice Questions & MCQ

1. Addition of Two Complex Numbers

z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the sum z₁ + z₂ is defined as 

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

For example,  z₁= (3 - 4i) and z₂ = (2 + 5i), then z₁ + z₂ is

(3 − 4i) + (2 + 5i) = (3 + 2) + (−4 + 5)i = 5 + i

2. Difference of Two Complex Numbers

z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the difference z₁ - z₂ is defined as 

z₁ - z₂ =  (a + ib) - (c + id) =  (a - c) + i(b - d)

For example,  z₁= (-5 + 7i) and z₂ = (-11 + 2i), then z₁ - z₂ is

(−5 + 7i) − (−11 + 2i) = −5 + 7i + 11 − 2i

                                  = −5 + 11 + 7i − 2i

                                  = (−5 + 11) + (7 − 2)i

                                  =  6 + 5i

3. Multiplication of Two Complex Numbers

z₁ = a + ib and z₂ = c + id be any two complex numbers. Then, the multiplication z₁・z₂ is defined as 

z₁・z₂ = (a + ib)・ (c + id)

          = ac + iad + ibc + i²bd

          = ac + i(ad + bc) - bd

          = (ac - bd) + i(ad + bc)

For example, z₁= (4 + 3i) and z₂ = (2 - 5i), then z₁・ z₂ is

(4 + 3i)(2 − 5i) = 4(2) − 4(5i) + 3i(2) − (3i)(5i)

                        =  8 − 20i + 6i − 15(i²)

                        =  (8 + 15) + (−20 + 6)i

                        = 23 - 14i

4. Division of Two Complex Numbers

z₁ = a + ib and z₂ = c + id (and z₂ is non-zero) be any two complex numbers. Then, the division    is defined as 

1. Closure law: sum of two complex number is a complex number, i.e. z₁ + z₂ is a complex number for all complex numbers z₁ and z₂.
2. Commutative law : for any two complex numbers  z₁ and z₂ ,  z₁ + z₂ = z₂ + z₁
3. Associative law : for any three complex numbers  z₁, z₂ and z₃ ,   (z₁ + z₂) + z₃ =   z₁ + (z₂ + z₃)
4. Additive identity: if the sum of a complex number z₁ with another complex number z₂ is z₁, then z₂ is called the additive identity.  We have z + 0 = z = 0 + z, so 0 is additive identity.
5. Additive inverse : To every complex number z = a + ib, we have the complex number – a + i(– b) (denoted as – z), called the additive inverse or negative of z. i.e. z + (-z) = 0 (-z is additive inverse).