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Properties of Complex Numbers

An imaginary number is a number that can be written as a + bi, where ‘a’ and ‘b’ are real numbers and b is not equal to 0. The imaginary numbers are simply a subset of the complex numbers, where we make no restrictions on ‘a’ and ‘b’.
Complex Numbers:
A complex number is a number that can be written as a + bi, where a and b are real numbers and $i = \sqrt{1}$. Complex numbers have two parts:
•    Real part
•    Complex part

From the definition of complex numbers, a complex number z = x + yi is defined by the two real numbers x and y. Hence, if we consider the real part x as the x coordinate in the rectangular coordinates system and the imaginary part y as the y coordinate then the complex number z can be represented by the point (x, y) on the plane. The number 0 is represented by the origin O.

The graphical representation of a complex number is shown as below:
Graphical Representation of Complex Numbers
where, the complex conjugate of z = x + yi is z bar = x - yi. It follows that the complex conjugate of x - yi is x + yi.Addition and Subtraction of 2 complex numbers is defined as follows:
z1 $\pm$ z2 = (x1 $\pm$ y1i) + (x2 $\pm$ y2i) = (x1 $\pm$ x2) + i( y1 $\pm$ y2)
Multiplication of 2 complex numbers is defined as follows:
z1z2 = (x1 + y1i) * (x2 + y2i) = x1x2 + y1x2i + x1y2i + y1y2i2
z1z2 = (x1x2 - y1y2) + (y1x2 + x1y2)i  ($\because$ i2 = -1 )

Properties of the complex numbers:

1). Commutative law for addition: z1 + z2 = z2 + z1.
2). Commutative law for multiplication: z1 * z2 = z2 * z1
3). Associative law for addition: z1 + (z2 + z3) = (z1 + z2) + z3
4. Associative law for multiplication: z1 * (z2 * z3) = (z1 * z2) * z3
5). Multiplication is distributive with respect to addition: z1 * (z2 + z3) = z1 * z2 + z1 * z3
6). If 2 complex numbers are multiplied and the result is zero, then it’s only possible if and only if at least one of the factors is zero.
7). Additive Inverses: z has a unique negative –z such that z + (–z) = 0. If z = x + yi, then –z = – x – yi.
8). Multiplicative Inverses: z = x + yi, provided z not equal to zero, will always has a unique inverse 1/z such that z (1/z) = 1. 1/z is known as the reciprocal of the complex number z. Here, 1/z= 1/(x + iy).
9). Additive Identity: There is a complex number v such that z + v = z for all complex numbers z. The number v is the ordered pair (0, 0).
10). Multiplicative Identity: There is a complex number e such that ze = z for all complex numbers z. The ordered pair (1, 0) = 1 + 0i is the unique complex number with this property.

Solved Examples:
Example 1: Represent the following complex numbers in a graph?
z = 3 + 2j
Solution: The complex number is represented as follows: 
Complex Numbers
Example 2: Simplify (2 + 3i) [(3 - 5i) (3 - 5i)]
Solution: (2 + 3i) (-16 - 30i)
=> (-32 - 48i - 60i - 90i2)
=> (-32 + 90 - 48i - 60i)
=> (58 - 108i)

Related Calculators
Adding Complex Numbers Calculator Complex Number Calculator
Dividing Complex Numbers Calculator Adding Complex Fractions Calculator

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