GMAT GRE 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th grade math 8th grade math 9th grade math 10th grade math 11th grade math 12th grade math Precalculus Worksheets Chapter wise Test MCQ's Math Dictionary Graph Dictionary Multiplicative tables Math Teasers NTSE Chinese Numbers CBSE Sample Papers |
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) = (ca-bd) + i(cb + da) But a, b, c , d are real numbers, so, ac - bd = ca - db and ad + bc = cb + da Hence, multiplication of complex-numbers is commutative. Associative Property : Consider the three complex numbers, z 1 = a + ib , z 2 = c + id and z 3 = e + if (z1 . z2 ). z3 = [(a + ib).( c + id )] .(e + if) =[(ac - bd)+ i(ad +bc)] . (e + if) =(ac-bd). e + i(ad +bc)e + i(ac -bd)f + i 2 (ad +bc).f =(ace -bde - adf -bcf) + i(ade + bce + acf -bdf) -------------(1) z1.(z2 . z3) = (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 22 = 9 - 4i 2) Find the multiplicative inverse of - 3 + 4i Let z = -3 + 4i , then z = z̄ /|z|
More To Explore
|
|||||||||||||||||