Here you will learn subtraction and addition of complex numbers and its properties with examples.
Let’s begin –
Addition of Complex Numbers
Let \(z_1\) = \(a_1 + ib_1\) and \(z_2\) = \(a_2 + ib_2\) be two complex numbers. Then their sum \(z_1 + z_2\) is defined as the complex number \((a_1 + a_2)\) + i\((b_1 + b_2)\).
It follows from the definition that the sum \(z_1 + z_2\) is a complex number such that
\(Re (z_1 + z_2)\) = \(Re (z_1)\) + \(Re (z_2)\) and
\(Im (z_1 + z_2)\) = \(Im (z_1)\) + \(Im (z_2)\)
Example : If \(z_1\) = 2 + 3i and \(z_2\) = 3 – 2i, then \(z_1 + z_2\) = (2 + 3) + (3 – 2)i = 5 + i
Properties :
(i) Addition is Commutative : For any two complex numbers \(z_1\) and \(z_2\), we have
\(z_1 + z_2\) = \(z_2 + z_1\)
(ii) Addition is Associative : For any three complex numbers \(z_1\), \(z_2\), \(z_3\), we have
(\(z_1\) + \(z_2\)) + \(z_3\) = \(z_1\) + (\(z_2\) + \(z_3\))
(iii) Existence of Additive Identity : The complex number 0 = 0 + i0 is the identity element for addition i.e. z + 0 = z = 0 + z for all z \(\in\) C.
(iv) Existence of Additive Inverse : For any complex number z = a + ib, there exists -z = (-a) + i(-b) such that z + (-z) = 0 = (-z) + z.
Subtraction of Complex Numbers
Let \(z_1\) = \(a_1 + ib_1\) and \(z_2\) = \(a_2 + ib_2\) be two complex numbers. Then the subtraction of \(z_2\) from \(z_1\) is denoted by \(z_1 – z_2\) and is defined as the addition of \(z_1\) and \(-z_2\).
Example : If \(z_1\) = -2 + 3i and \(z_2\) = 4 + 5i, then \(z_1 – z_2\) = (-2 + 3i) + (-4 – 5i) = (-2 – 4) + i(3 – 5) = -6 – 2i
