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 

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