We know that the solution of the quadratic equation

ax^{2} + bx + c = 0, a≠0

are given by quadratic formula

x = -b ± √(b^{2}-4ac)/2a

Now when b^{2}-4ac < 0 the equation cannot be solved in real number system. For this solution, we need the set of complex numbers.

i= √(-1)

The set of complex numbers denoted by ℂ is defined as:

ℂ = {a+bi∶a,b ∈ **R**,i = √(-1)}

The notation a + bi is called writing **complex number in standard form**

a is called the real part of complex number z= a + bi (also written as a = Re z) and b is called the imaginary part of complex number z = a+bi (also written as b = Im z ).

Examples of complex numbers can be given as 2+4i or √(-9 )=9i etc.

Two complex numbers z_{1}= a_{1}+b_{1} i and z_{2}= a_{2}+ b_{2} i are equal if their real and imaginary parts are equal, which means a_{1}=a_{2} and b_{1}= b_{2}.

Every real number is a complex number with its imaginary part 0. Thus a = a + 0i

So R ⊂ ℂ

The complex number z̅= a - bi is called the conjugate of a complex number z = a + bi .

The absolute value of complex number z is defined as |z|= √(a^{2}+b^{2})

Two complex numbers can be added, subtracted, multiplied and divided resulting in new complex numbers.

If z_{1}= a_{1} + b_{1} i and z_{2} = a_{2} + b_{2} i then

z_{1} + z_{2} = (a_{1} + a_{2} ) + (b_{1} + b_{2} )i

z_{1} - z_{2} = (a_{1} - a_{2} )+(b_1-b_2 )i

z_{1} z_{2} = (a_{1} a_{2} - b_{1} b_{2} )+(a_{1} b_{2} + a_{2} b_{1} )i

z_{1}/z_{2} = a_{1} + b_{1} i/a_{2} + b_{2} i = (a_{1} + b_{1} i)( a_{2} - b_{2} i) )/(a_{2} + b_{2} i)(a_{2}- b_{2} i) = (a_{1} a_{2}+ b_{1} b_{2})/(a_{2}^{2} + b_{2}^{2} ) - (a_{1} b_{2} + a_{2} b_{1})/(a_{2}^{2} + b_{2}^{2} ) i

Then,

1/z= 1/(a+bi)= (a-bi)/((a+bi)(a-bi)) = (a-bi)/(a^{2}+b^{2} )= a/(a^{2+b2 ) - b/(a2+b2 ) i}

For simplifying complex numbers we must remember the various powers of imaginary number I, especially while multiplying imaginary numbers:

- i
^{2}= 1 , - i
^{3}= -i , - i
^{4}= 1 , - i
^{5}= i , - i
^{6}= -1 , - i
^{7}= -i…

- or the conjugate of the sum of complex numbers is the sum of the conjugates of complex number
- z̅
_{1}̅z̅_{2}̅ = z̅_{1}.z̅_{2}or the conjugate of the product of complex numbers is the product of the conjugates of complex number - z̅. z = |z|
^{2}or the product of the complex number with it’s own conjugate is equal to the square of the absolute value of that complex number - z̅ + z = 2 Re z or the addition of a complex number with its own conjugate is equal to twice the real part of the complex number
- z - z̅ =2i Im z or subtracting the conjugate of a complex number from the complex number is equal to two times iota multiplied by the imaginary part of the complex number
- Re z ≤ |z| or the real part of the complex number is less than or equal to the absolute value of complex number
- z̿ = z or the conjugate of a conjugate of complex number is the complex number itself.
- z is real if and only if z̅ = z or if the conjugate of a complex number is equal to the complex number itself then the complex number is a real number, that is it has only a real part and no imaginary part.

Let z_{1} = 3 - 2i and z_{2} = -2 + 3i then

Re z_{1} = 3, Im z_{1} = -2 and

Re z_{2} = -2, Im z_{2} = 3

|z_{1} | = √(3^{2} + (-2)^{2} ) = √13

|z_{2} | = √((-2)^{2}+ 3^{2} )= √13

z_{1} + z_{2} = (3-2)+(-2+3)i = 1+i

z_{1} -z_{2} = (3 -( -2)) + (-2 -3)i = 5-5i

z_{1} z_{2} = (3*(-2)-(-2*3))+(3*3+(-2*-2))i = 13i

z_{1}/z_{2} = 3-2i/-2+ 3i = 3*(-2) + (-2*3)/4+9 - 3*3 +(-2 * -2))/4+9 i = -12/13 + 13/13 i= -12/13 + i

1/z_{1} = 1/3-2i = 3/13 + 2/13 i

1/z_{2} = 1/-2+3i = -2/13 - 3/13 i

z̅_{1} =3 +2i

z̅_{2} = -2 -3i

z̅_{1} + z̅_{2} = (3 - 2)+(2 - 3)i = 1-i

**Hence verified**:

z̅_{1} . z̅_{2} = (3*(-2)-(-2*3)) + (3*-3+(2*-2))i = -13i

z̅_{1}̅z̅_{2}̅ = 1̅3̅i = - 13i

**Hence verified**: z̅_{1}̅z̅_{2}̅ = z̅_{1} . z̅_{2}

z̅_{1}.z_{1} = (3-2i)*(3+2i)=(9+4)+(-2*3+3*2)i=13=(√13)^{2} = |z_{1} |^{2}

**Hence verified**: z̅.z = |z|^{2}

z̅_{1} + z_{1} = (3-2i)+(3+2i)= 6 = 2*3 = 2Re z_{1}

**Hence verified: z+z̅ = 2 Re z**

z_{1} - z_{1}̅ =(3-2i) - (3+2i) = -4i = 2i(-2) = 2i Im z_{1}

**Hence verified: z-z̅ = 2i Im z**

**Hence verified: Re z ≤ |z|**

z̿_{1} = 3̿-̿2̿i̿ = (3+2i) ̅ = 3-2i = z_{1}

**Hence verified: z̿ = z**

Complex numbers have vast applications in the field of electrical engineering, study of electrical circuits, trigonometry, calculus, quantum mechanics, and study of waves such as electricity, light, and sound. Or simply in finding square root of negative numbers.

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