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.
Dec 04, 23 02:14 PM
Dec 04, 23 01:50 PM
Dec 04, 23 01:49 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.