Here you will learn multiplication of complex numbers and its properties with examples.
Letโs begin โ
Multiplication of Complex Numbers
Let \(z_1\) = \(a_1 + ib_1\) and \(z_2\) = \(a_2 + ib_2\) be two complex numbers. Then the multiplication of \(z_1\) with \(z_2\) is denoted by \(z_1 z_2\) and is defined as the complex number
\((a_1a_2 โ b_1b_2)\) + i\((a_1b_2 + a_2b_1)\).
Thus,ย \(z_1\)\(z_2\) = \(a_1 + ib_1\)\(a_2 + ib_2\)
\(\implies\)ย \(z_1\)\(z_2\) = \((a_1a_2 โ b_1b_2)\) + i\((a_1b_2 + a_2b_1)\)
\(\implies\)ย \((a_1a_2 โ b_1b_2)\) = [\(Re(z_1) Re(z_2) โ Im(z_1) Im(z_2)\)] + i [\(Re(z_1) Im(z_2) + Re(z_2) Im(z_1)\)]
Example : If \(z_1\) = 3 + 2i and \(z_2\) = 2 โ 3i, then
\(z_1 z_2\) = (3 + 2i)(2 โ 3i)
= \((3 \times 2 โ 2 \times (-3)) + i(3 \times -3 + 2 \times 2)\) = 12 โ 5i
Note : The product \(z_1z_2\) can also be obtained if we actually carry out the multiplication (\(a_1 + ib_1\))(\(a_2 + ib_2\)) as given below :
(\(a_1 + ib_1\))(\(a_2 + ib_2\)) = \(a_1a_2\) + \(ia_1b_2\) + \(ib_1a_2\) + \(i^2b_1b_2\)
= \((a_1a_2 โ b_1b_2)\) + i\((a_1b_2 + a_2b_1)\)ย ย ย ย ย ย [\(because\)ย \(i^2\) = -1]
Properties of Multiplication
(i) Multiplication is Commutative : For any two complex numbers \(z_1\) and \(z_2\), we have
\(z_1 z_2\) = \(z_2 z_1\)
(ii) Multiplication is Associative : For any three complex numbers \(z_1\), \(z_2\), \(z_3\), we have
(\(z_1\) \(z_2\)) \(z_3\)ย = \(z_1\) (\(z_2\) \(z_3\))
(iii) Existence of Identity Element for Multiplicationย : The complex number 1 = 1 + i0 is the identity element for multiplication i.e. for every complex number z, we have
z.1 = z = 1.z
(iv) Existence of Multiplicative Inverse : Corresponding to every non-zero complex number z = a + ib there exists a complex number \(z_1\) = x + iy such that
\(z.z_1\) = 1 = \(z_1.z\)
(v) Multiplications of complex numbers is distributive over addition of complex numbers : For any three complex numbers \(z_1\), \(z_2\), \(z_3\), we have
(i) \(z_1(z_2 + z_3)\) = \(z_1z_2 + z_1z_3\)
(ii) \((z_2 + z_3)z_1\) = \(z_2z_1 + z_3z_1\)
