Performing Algebraic Operations on Complex Numbers

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
The real part of the complex number is represented by ‘a’ and the imaginary part is represented by ‘b’.
The imaginary unit ‘i’ is defined as the square root of -1: i.e., i^2 = -1.
The complex conjugate of a complex number a + bi is a – bi. By changing the sign of the imaginary part we get the conjugate.

To add or subtract complex numbers, combine like terms. This means, add or subtract the real parts together, and the imaginary parts together.

Multiplying complex numbers involves applying the distributive property (also known as the FOIL method in binomials), and using the definition of i^2 = -1.

Dividing complex numbers requires the use of the conjugate of the denominator. Multiply the numerator and denominator by the conjugate of the denominator, simplify, then separate into the real and imaginary parts if necessary.

Complex numbers follow the commutative, associative, and distributive laws of real numbers in addition and multiplication.
That is, for all complex numbers x, y, and z, the following hold:
- Commutative laws: x + y = y + x and xy = yx
- Associative laws: (x + y) + z = x + (y + z) and (xy)z = x(yz)
- Distributive laws: x(y + z) = xy + xz

The operations on complex numbers are extensively applied in fields such as engineering, physics, and computer science where they facilitate solutions of certain types of equations that have no real solutions.

Similar rules to real number arithmetic apply when performing algebraic operations on complex numbers.
Always remember that i^2 is equal to -1. This is especially useful when multiplying and dividing complex numbers.
Using the conjugate is a crucial step in complex number division.
Practice regularly to gain confidence in combining and manipulating complex numbers.