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### Argument

The argument of a complex number gives the angle with the *x*-axis

##### Explanation

The complex number *z* can be written as

where |*z*| is a positive real number, and you call this the modulus of *z*. The Greek letter *θ* (theta) is a real number that specifies the angle with the *x*-axis, and is called the argument. It is also called the phase or amplitude. Thus *x* = |*z*| cos *θ* and *y* = |*z*| sin *θ*.

The argument of *z* in the interval [0, 2π) is called the principle value. As principle value also the interval (−π, π] can be selected. For a point on the *y*-axis, where *x* = 0, applies

The complex argument is calculated as

Special values of the complex argument are

From the definition follows that the product of two complex numbers (*z* ≠ 0) is equal to the sum of their arguments

It follows

with as special case

A division of two complex numbers gives

##### Example 1

##### Example 2

With arg (*z*) and modulus |*z*| the complex number *z* can be written as

##### Example 3

If *z* is not a pure imaginary number, so is not on the vertical *y*-axis, it applies

##### Details

It applies *θ* = arccis *x* = arg* x*