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Multiplication of Complex NumbersThe multiplication of complex numbers: The product of two complex numbers z _{1} = a +ib and z _{2} = c + id is defined as a complex number obtained by the multiplication of these two numbers as binomial governed by the rules of algebra dn substituting 1 for i ^{2} . We have,z _{1} z _{2} = (a + ib)(c + id) = ac + i ad + i bc + i ^{2} bd = (ac  bd) + i(ad + bc) For multiplication of complex numbers, the students should know the values of different powers of 'i' . The values of different powers of 'i' are given below.
For Example : i ^{17} = i ^{16} i = i because i ^{16} is same as i ^{4} In general, we can say that for any integer ‘k’
Properties of multiplication of complex numbersClosure : The product of two complex numbers is , by definition , a complex number. Hence, the set of complex numbers is closed under multiplication.Commutative property : For two complex numbers z _{1} = a + ib and z _{2} = c + id , we have z _{1} . z _{2} = (a + ib)(c + id) = (ac bd) + i(ad + bc) (since i ^{1} = 1 z _{2} . z _{1} = (c + id)(a + ib) = (cabd) + i(cb + da) But a, b, c , d are real numbers, so, ac  bd = ca  db and ad + bc = cb + da Hence, multiplication of complexnumbers is commutative. Associative Property : Consider the three complex numbers, z _{1} = a + ib , z _{2} = c + id and z _{3} = e + if (z_{1} . z_{2} ). z_{3} = [(a + ib).( c + id )] .(e + if) =[(ac  bd)+ i(ad +bc)] . (e + if) =(acbd). e + i(ad +bc)e + i(ac bd)f + i ^{2} (ad +bc).f =(ace bde  adf bcf) + i(ade + bce + acf bdf) (1) z_{1}.(z_{2} . z_{3}) = (a+ib).[(c +id).(e +if)] = (ace  adf  bcf bde) + i(acf + ade + bce bdf) (2) Thus, from (1) and (2) (z _{1} . z _{2} ). z _{3} = z _{1} .(z _{2} . z _{3} ) Multiplication Identity: Let c + id be the multiplicative identity of a + ib. Then (a + ib)(c + id) = a + ib ⇒ (ac  bd) + i(ad + bc) = a + ib ⇒ ac  bd = a and ad + bc = b ac  a = bd and bc  b = ad a(c  1) = bd (1) b(c  1) = ad (2) Multiply equation (1) by a and equation(2) by b and then add a ^{2} (c  1) = abd b ^{2} (c  1) = abd  (a ^{2} + b ^{2} )(c  1) = 0 so either a ^{2} + b ^{2} = 0 or c  1 = 0 but a ^{2} + b ^{2} ≠ 0 so c  1 = 0 ⇒ c = 1 ∴ d = 0 c + id = 1 + i0 = 1 Hence the multiplicative identity of the complex number is 1. Multiplicative inverse: A complex number 'w' is called the multiplicative inverse of complex number z, if z . w = 1. The multiplicative inverse is denoted by z FOIL ( 4 x 2)(4 x 12i)(2i x 2)(2i x 12i) = + + + 8(24) =  16 + 52 i ^{2}^{2} = 9  4i 2) Find the multiplicative inverse of  3 + 4i Let z = 3 + 4i , then z = z̄ /z
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