# Discuss Discuss arithmetic of complex number. Is it special?

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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No, it is not special, as long as you keep in mind that the complex numbers have the following form, called rectangular form:

z = x + iy, where x is called the real part of the number and y is called the imaginary part of the complex number.

Also, you should know that i is the square root of -1.

The 4 basic arithmetic operations are:

We'll treat "i" as a variable and we'll combine the real parts and imaginary parts together.

z1 = x1 + i*y1

z2 = x2 + i*y2

z1 + z2 = (x1 + x2) + i*(y1 + y2)

2) subtraction

It is happening like in addition case, but we have to pay attention to the signs:

z1 - z2 = (x1 - x2) + i(y1 - y2)

3) multiplication

We'll have to keep in mind that i^2 is turning into -1.

z1*z2 = (x1 + iy1)(x2 + iy2)

z1*z2 = x1*x2 + i*x1y2 + i*x2y1 + y1y2*i^2

z1*z2 = (x1x2 - y1y2) + i(x1y2 + x2y1)

4) division

Any division between 2 complex number has the following algorithm:

-firsat, we'll have to multiply the numerator by the conjugate of denominator, since we are not allowed to keep complex numbers to the denominator.

-then, we can separate the real part and the imaginary part.

z1/z2 = z1*z2'/z2*z2'