Operations on complex numbers | Part-1

Addition and Subtraction of Complex Numbers

A complex number can be written as \( z = a + ib \), where \( a \) denotes the real part (\( \mathrm{Re}(z) \)) of the number and \( b \) represents the imaginary component (\( \mathrm{Im}(z) \)), with \( i \) being the imaginary unit defined by \( i = sqrt{-1} \)

Addition of Complex Numbers

To add two complex numbers \( z_1 = a + ib \) and \( z_2 = c + id \), we add the real parts and imaginary parts separately:

\( z_1 + z_2 = (a + c) + i(b + d) \)

Example:
Take the complex numbers \( z_1 = 3 + 4i \) and \( z_2 = 2 - i \). Their sum can be calculated in the following way: \( z_1 + z_2 = (3 + 2) + i(4 + (-1)) = 5 + 3i \)

Subtraction of Complex Numbers

Subtracting \( z_2 \) from \( z_1 \) involves taking the difference of their real parts and their imaginary parts individually.

\( z_1 - z_2 = (a - c) + i(b - d) \)

Example:
Let \( z_1 = 5 + 7i \) and \( z_2 = 3 + 2i \), then: \( z_1 - z_2 = (5 - 3) + i(7 - 2) = 2 + 5i \)

Geometric Interpretation

In the complex plane, each complex number is represented as a point \( (a, b) \). Adding or subtracting complex numbers follows the same rules as combining or separating vectors in the plane. Adding two complex numbers \( z_1 \) and \( z_2 \) results in a sum that corresponds to the diagonal of the parallelogram formed by the position vectors \( z_1 \) and \( z_2 \) within the complex plane.

Key Points

• Addition and subtraction are done by combining real parts and imaginary parts separately.
• \( z_1 + z_2 = z_2 + z_1 \) (commutative law)
• \( (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \) (associative law)
• \( z + (-z) = 0 \) (additive inverse)

Understanding how to add and subtract complex numbers is essential before learning about multiplication, division, and other advanced operations in complex number theory.