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# Complex number

A complex number is defined as a pair of real numbers $\displaystyle{ z = (x, y) }$ where the following operations are defined:

• Addition: $\displaystyle{ z_1 + z_2 = (x_1 + x_2, y_1 + y_2) }$
• Multiplication: $\displaystyle{ z_1 z_2 = (x_1 y_1 - x_2 y_2, x_2 y_1 + x_2 y_1) }$

When the second element equals zero the complex numbers behaves as real numbers. That's why the first element of the complex number is known as the real part and the second element as the imaginary part.

Multiplying $\displaystyle{ (0, 1) (0, 1) }$ we get $\displaystyle{ (-1, 0) }$. Since no real number is the square root of -1, we can now understand why the second element is the imaginary part.

An alternate (and more used) notation is $\displaystyle{ z = x + iy }$. From the previous paragraph we get: $\displaystyle{ i^2 = -1 }$.

Using this notation and the definitions above we can deduce all basic operations on complex numbers:

$\displaystyle{ z_1 + z_2 = (x_1 + x_2) + (y_1 + y_2) i }$
$\displaystyle{ z_1 - z_2 = (x_1 + x_2) + (-y_1 - y_2) i }$
$\displaystyle{ z_1 z_2 = (x_1 y_1 - x_2 y_2) + (x_1 y_2 + x_2 y_1) i }$
$\displaystyle{ \frac {z_1}{z_2} = \frac {x_1 y_1 + x_2 y_2}{x_2^2 + y_2^2}\,+\,\frac {x_2 y_1 - x_1 y_2}{x_2^2 + y_2^2} \,i }$
$\displaystyle{ \sqrt{x+iy} = \sqrt{\frac{\left|x+iy\right| + x}{2}} \pm i \sqrt{\frac{\left|x+iy\right| - x}{2}} }$
$\displaystyle{ \left|x+iy\right| = \sqrt{x^2+y^2} }$