Here we will discuss about the commutative property of multiplication of complex numbers.
Commutative property of multiplication of two complex numbers:
For any two complex number z\(_{1}\) and z\(_{2}\), we have z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\).
Proof:
Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is, where p, q, r and s are real numbers. Them
z\(_{1}\)z\(_{2}\) = (p + iq)(r + is) = (pr  qs) + i(ps  rq)
and z\(_{2}\)z\(_{1}\) = (r + is) (p + iq) = (rp  sq) + i(sp  qr)
= (pr  qs) + i(ps  rq), [Using the commutative of multiplication of real numbers]
Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)
Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z\(_{2}\) ϵ C.
Hence, the multiplication of complex numbers is commutative on C.
Examples on commutative property of multiplication of two complex numbers:
1. Show that multiplication of two complex numbers (2 + 3i) and (3 + 4i) is commutative.
Solution:
Let, z\(_{1}\) = (2 + 3i) and z\(_{2}\) = (3 + 4i)
Now, z\(_{1}\)z\(_{2}\) = (2 + 3i)(3 + 4i)
= (2 ∙ 3  3 ∙ 4) + (2 ∙ 4 + 3 ∙ 3)i
= (6  12) + (8 + 9)i
=  6 + 17i
Again, z\(_{2}\)z\(_{1}\) = (3 + 4i)(2 + 3i)
= (3 ∙ 2  4 ∙ 3) + (3 ∙ 3 + 2 ∙ 4)i
= (6  12) + (9 + 8)i
= 6 + 17i
Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)
Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z2 ϵ C.
Hence, the multiplication of two complex numbers (2 + 3i) and (3 + 4i) is commutative.
2. Show that multiplication of two complex numbers (3  2i) and (5 + 4i) is commutative.
Solution:
Let, z\(_{1}\) = (3  2i) and z\(_{2}\) = (5 + 4i)
Now, z\(_{1}\)z\(_{2}\) = (3  2i)(5 + 4i)
= (3 ∙ (5)  (2) ∙ 4) + ((2) ∙ 4 + (5) ∙ (2))i
= (15  (8)) + ((8) + 10)i
= (15 + 8) + (8 + 10)i
=  7 + 2i
Again, z\(_{2}\)z\(_{1}\) = (5 + 4i)(3  2i)
= ((5) ∙ 3  4 ∙ (2)) + (4 ∙ 3 + (2) ∙ 4)i
= (15 + 8) + (12  8)i
= 7 + 2i
Therefore, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\)
Thus, z\(_{1}\)z\(_{2}\) = z\(_{2}\)z\(_{1}\) for all z\(_{1}\), z\(_{2}\) ϵ C.
Hence, the multiplication of two complex numbers (3  2i) and (5 + 4i) is commutative.
11 and 12 Grade Math
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