# Complex numbers operations

Add, subtract, multiply and divide complex numbers.
Allowed: constants, operators and i. To multiply use a*b not ab
Allowed: constants, operators and i. To multiply use a*b not ab

Assume that z_1 and z_2 are two complex numbers such as,

z_1= a_1 +b_1.i
z_2= a_2 +b_2.i

Then,

z_1 +z_2 = (a_1 + a_2) + (b_1 +b_2) * i

This means that to add two complex numbers, just add their real parts and imaginary parts separately.

Example:

z_1= 4 + i
z_2= -1 - 3*i

z_1 +z_2 = (4 -1) + (1 -3) * i = 3 -2*i

## Subtraction of complex numbers

Let z_1 and z_2 be two complex numbers such as,

z_1= a_1 +b_1.i
z_2= a_2 +b_2.i

Then,

z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2) * i

So, to subtract two complex numbers, just subtract their real parts and their imaginary parts.

Example:

z_1= i
z_2= -6 - 3*i

z_1 - z_2 = (0 -(-6)) + (1 -(-3)) * i = 6 + 4*i

## Multiplication of complex numbers

If z_1 and z_2 are two complex numbers such as then,

z_1= a_1 +b_1.i
z_2= a_2 +b_2.i

then,

z_1 * z_2 = (a_1 +b_1*i) * (a_2 +b_2*i)

z_1 * z_2 = a_1 * a_2 +a_1*b_2*i + b_1*a_2*i + b_1*b_2*i^2

We know that i^2 = -1 , so,

z_1 * z_2 = a_1*a_2-b_1*b_2 + i*(a_1*b_2+b_1*a_2)

Exemple:

z_1= 1 + 2*i
z_2= 5 - 4*i

z_1 * z_2 = (1 +2*i) * (5 -4*i)
z_1 * z_2 = 5 -4*i +(2*i)*5-2*i*4*i
z_1 * z_2 = 5 -4*i +10*i -8*i^2
z_1 * z_2 = 5 -4*i + 10*i + 8
z_1 * z_2 = 13 + 6*i

## Division of complex numbers

z_1 and z_2 are two complex numbers such as,

z_1= a_1 +b_1.i
z_2= a_2 +b_2.i

Then,

\frac{z_1}{z_2} = \frac{a_1 +b_1*i}{a_2 +b_2*i}

We use the conjugate of z_2 to get rid of i in the denominator (See more details here Conjugate of a complex number):

bar z_2 = a_2 -b_2.i

We use this identity,

z_2 * bar z_2 = (a_2 +b_2.i) * (a_2 -b_2.i) = a_2^2 + b_2^2

\frac{z_1}{z_2} = \frac{(a_1 +b_1.i)*bar z_2}{(a_2 +b_2.i)*bar z_2} = \frac{(a_1 +b_1.i)(a_2 -b_2.i)}{(a_2 +b_2.i)*(a_2 -b_2.i)} = \frac{(a_1*a_2 +b_1*b_2) + (b_1*a_2 - a_1*b_2)*i}{(a_2^2 +b_2^2)}

Exemple:

z_1= 3 -i
z_2= 2 - 5*i

\frac{z_1}{z_2} = \frac{3-i}{2-5*i}

We multiply the numerator and the denominator by the conjugate of z_2:
bar z_2 = 2+5*i

\frac{z_1}{z_2} = \frac{(3-i)*(2+5*i)}{(2-5*i)(2+5*i)} = \frac{6+15*i-2*i+5}{4+25} = \frac{11+13*i}{29} = \11/29+13/29*i