# 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.

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