Associative Property of Multiplication of Complex Numbers

Here we will discuss about the associative property of multiplication of complex numbers.

Commutative property of multiplication complex numbers:

For any three complex numbers z\(_{1}\), z\(_{2}\) and z\(_{3}\), we have (z\(_{1}\)z\(_{2}\))z\(_{3}\) = z\(_{1}\)(z\(_{2}\)z\(_{3}\)).

Proof:

Let z\(_{1}\) = a + ib, z\(_{2}\) = c + id and z\(_{3}\) = e + if be any three complex numbers.

Then (z\(_{1}\)z\(_{2}\))z\(_{3}\) = {(a + ib)(c + id)}(e + if)

                    = {(ac - bd) +i(ad + cb)}(e + if)

                    = {(ac - bd)e - (ad + cb)f) + i{(ac - bd)f + (ad + cb)e)

                    = {a(ce - df) - b(cf + ed)} + i{b(ce - df) + a(ed + cf)

                    = (a + ib){(cf - df) + i(cf + ed)}

                    = z\(_{1}\)(z\(_{2}\)z\(_{3}\))

Thus, (z\(_{1}\)z\(_{2}\))z\(_{3}\) = z\(_{1}\)(z\(_{2}\)z\(_{3}\)) for all z\(_{1}\), z\(_{2}\), z\(_{3}\) ϵ C.

Hence, multiplication of complex numbers is associative on C.

 

Solved example on commutative property of multiplication of complex numbers:

Show that multiplication of complex numbers (2 + 3i), (4 + 5i) and (1 + i) is associative.

Solution:

Let z\(_{1}\) = (2 + 3i), z\(_{2}\) = (4 + 5i) and z\(_{3}\) = (1 + i)

Then (z\(_{1}\)z\(_{2}\))z\(_{3}\) = {(2 + 3i)(4 + 5i)}(1 + i)

                    = (2 4 - 3 5) + i(2 5 + 4 3)}(1 + i)

                    = (8 - 15) + i(10 + 12)}(1 + i)

                    = (-7 + 22i)(1 + i)

                    = (-7 1 - 22 1) + i(-7 1 + 1 22)

                    = (-7 – 22) + i(-7 + 22)

                    = -29 + 15i

Now, z\(_{1}\)(z\(_{2}\)z\(_{3}\)) = (2 + 3i){(4 + 5i)(1 + i)}

                    = (2 + 3i){(4 1 - 5 1) + i(4 1 + 1 5)}

                    = (2 + 3i){(4 - 5) + i(4 + 5)}

                    = (2 + 3i)(-1 + 9i)

                    = {2 (-1) - 3 9} + i{2 9 + (-1) 3}

                    = (-2 - 27) + i(18 - 3)

                    = -29 + 15i

Thus, (z\(_{1}\)z\(_{2}\))z\(_{3}\) = z\(_{1}\)(z\(_{2}\)z\(_{3}\)) for all z\(_{1}\), z\(_{2}\), z\(_{3}\) ϵ C.

Hence, multiplication of complex numbers (2 + 3i), (4 + 5i) and (1 + i) is associative.





11 and 12 Grade Math 

From Associative Property of Multiplication of Complex Numbers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.