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# Complex Numbers

work in progress

## Definition

Real numbers have a property specifying that for any real number r,

r ² > 0

Complex number is an extention of the real numbers, adding a number i satisfying the equation i ^ 2 = -1

A complex number is usually written in the form: a + bi

a and b are real numbers, a being call the real part, and b the imaginary part.

Two complex numbers a + bi and c + di are considered equal if :
a=c AND b=d

## Operations

Adding two complex numbers is simply done by adding the real parts together and the imaginary parts together.

(a+bi) + (c+di) = (a+b) + (c+d)i

+ (Complex)

### Subtraction

Subtracting two complex numbers is simply done by adding the real parts together and the imaginary parts together.

(a+bi) - (c+di) = (a-b) + (c-d)i

- (Complex)

### Multiplication

To process multiplication, we use standard algebra rules:

(a + bi) * (c + di) = (a*c - b*d) + (a*d + b*c)i

decomposition:
(a + bi) * (c + di) = a*c + a*di + c * bi + bi * di = a*c + (a*d + b*c)i + b*d i² = (a*c - b*d) + (a*d + b*c)i

* (Complex)

### Division

(a + bi) / (c + di) = ( (ac + bd + (bc - ad)i ) / (c² + d²)

/ (Complex)

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