Complex Number

COMPLEX NUMBERS
- A complex number is written in the form of a + bi where a and b are real numbers.
- a is called real part and bi is called imaginary part
- i = and i^2 = -1
- Generally, ( -1 )^even no. = 1
( -1 )^odd no. = -1

For the quadratic equation, ax^2 + bx + c =0, we are use the formula below to solve the equation

Addition and Subtraction of Complex Numbers

If z = x + yi and w = u + vi,

thus,
z + w = x + yi + u + vi
= (x + u) + (y + v)i
z – w = x + yi - u + vi
= (x - u) + (y - v)i

Multiplication of Complex Numbers

i) If z = x + yi and w = u + vi,

thus,

zw = (x + yi)( u + vi)
= (x + yi)(u) + (x + yi)(vi)

ii) If z = x + yi and w = x – yi,

thus

zw = ( x + iy) ( x – iy )
= x ^2 - (yi)^2
= x^ 2 + y^2 ( real number)

So, w is known as complex conjugate of z


Division of Complex Numbers

i. If z = x + yi and w = u + vi

thus,


where u – vi is conjugate of w

ii. For the division process, the denominator must be a real number


Equality of Complex Numbers

i) Let say z = x + yi and w = u + vi where z = w

thus,

x + yi = u + vi
x – u = (v – y)i

ii) Therefore, x + yi = u + vi if and only if x = u and y = v