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:**

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:**

z_{1} $\pm$ z_{2} = (x_{1} $\pm$ y_{1}i) + (x_{2} $\pm$ y_{2}i) = (x_{1} $\pm$ x_{2}) + i( y_{1} $\pm$ y_{2})**Multiplication of 2 complex numbers is defined as follows:**

z_{1}z_{2} = (x_{1} + y_{1}i) * (x_{2} + y_{2}i) = x_{1}x_{2} + y_{1}x_{2}i + x_{1}y_{2}i + y_{1}y_{2}i^{2}

z_{1}z_{2} = (x_{1}x_{2} - y_{1}y_{2}) + (y_{1}x_{2} + x_{1}y_{2})i ($\because$ i^{2} = -1 )

Properties of the complex numbers:**1)**. Commutative law for addition: z_{1} + z_{2} = z_{2} + z_{1}.**2)**. Commutative law for multiplication: z_{1} * z_{2} = z_{2} * z_{1}**3)**. Associative law for addition: z_{1} + (z_{2} + z_{3}) = (z_{1} + z_{2}) + z_{3}

4. Associative law for multiplication: z_{1} * (z_{2} * z_{3}) = (z_{1} * z_{2}) * z_{3}**5)**. Multiplication is distributive with respect to addition: z_{1} * (z_{2} + z_{3}) = z_{1} * z_{2} + z_{1} * z_{3}**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:

Example 2: Simplify (2 + 3i) [(3 - 5i) (3 - 5i)]

Solution: (2 + 3i) (-16 - 30i)

=> (-32 - 48i - 60i - 90i^{2})

=> (-32 + 90 - 48i - 60i)

=> (58 - 108i)

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