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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

 


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