Multiplication of Complex Numbers
The 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.
| i | √ -1 |
| i2 | -1 |
| i3 | - i |
| i4 | 1 |
| i5 | i = √ -1 |
| i6 | -1 |
| i7 | -i |
| i8 | 1 |
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’
|
i4k = 1 , i i4k +1 = i i4k+ 2 = -1 , i4k+3 = -i |
Properties of multiplication of complex numbers
Closure : 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|
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