Argument (complex analysis)

<h2 id="definition">Definition</h2>

An argument of the nonzero complex number z = x + iy, denoted arg(z), is defined in two equivalent ways:

<ol><li>Geometrically, in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>, as the <a href="/facts/2D_polar_angle/HBko4YoN">2D polar angle</a> 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 from the positive real axis to the vector representing z. The numeric value is given by the angle in <a href="/facts/Radian/Srd38Sxa">radians</a>, and is positive if measured counterclockwise.</li>
<li>Algebraically, as any real quantity 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 such that 
 
 
 
 z
 =
 r
 (
 cos
 ⁡
 φ
 +
 i
 sin
 ⁡
 φ
 )
 =
 r
 
 e
 
 i
 φ
 
 
 
 
 {\displaystyle z=r(\cos \varphi +i\sin \varphi )=re^{i\varphi }}
 
 for some positive real r (see <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a>). The quantity r is the <a href="/facts/Absolute_value/dwJBpEs5">modulus</a> (or absolute value) of z, denoted |z|: 
 
 
 
 r
 =
 
 
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 
 
 .
 
 
 {\displaystyle r={\sqrt {x^{2}+y^{2}}}.}
 
</li></ol>
The argument of zero is usually left undefined. The names <a href="/facts/Magnitude_(mathematics)/KCn9PoF0">magnitude</a>, for the modulus, and <a href="/facts/Phase_(waves)/cpGtuVll">phase</a>,<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> for the argument, are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any non-zero complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2π <a href="/facts/Radian/Srd38Sxa">radians</a> (a complete <a href="/facts/Turn_(angle)/QczxAyuM">turn</a>) are the same, as reflected by figure 2 on the right. Similarly, from the <a href="/facts/Periodic_function/Sm8lJQsF">periodicity</a> of <a href="/facts/Sine/gQ6LXK8z">sin 
</a> and <a href="/facts/Cosine/czej7ur0">cos</a>, the second definition also has this property.

<h2 id="principal-value">Principal value</h2>

Because a complete rotation around the origin leaves a complex number unchanged, there are many choices which could be made for 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
 by circling the origin any number of times. This is shown in figure 2, a representation of the <a href="/facts/Multi-valued/WH5ua6BL">multi-valued</a> (set-valued) function 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 arg
 ⁡
 (
 x
 +
 i
 y
 )
 
 
 {\displaystyle f(x,y)=\arg(x+iy)}
 
, where a vertical line (not shown in the figure) cuts the surface at heights representing all the possible choices of angle for that point.
When a <a href="/facts/Well-defined/GAc0XbWz">well-defined</a> function is required, then the usual choice, known as the <a href="/facts/Principal_value/bCHR0LmD">principal value</a>, is the value in the open-closed <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> (−π, π] radians, that is from −π to π <a href="/facts/Radian/Srd38Sxa">radians</a> excluding −π radians itself (equiv., from −180 to +180 <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a>, excluding −180° itself). This represents an angle of up to half a complete circle from the positive real axis in either direction.
Some authors define the range of the principal value as being in the closed-open interval [0, 2π).

<h3>Notation</h3>
The principal value sometimes has the initial letter capitalized, as in Arg z, especially when a general version of the argument is also being considered. Note that notation varies, so arg and Arg may be interchanged in different texts.
The set of all possible values of the argument can be written in terms of Arg as:

arg
        ⁡
        (
        z
        )
        =
        {
        Arg
        ⁡
        (
        z
        )
        +
        2
        π
        n
        ∣
        n
        ∈
        
          Z
        
        }
        .
      
    
    {\displaystyle \arg(z)=\{\operatorname {Arg} (z)+2\pi n\mid n\in \mathbb {Z} \}.}

<h2 id="computing-from-the-real-and-imaginary-part">Computing from the real and imaginary part</h2>
Main article: <a href="/facts/Atan2/0JDnUYfP">atan2</a>
If a complex number is known in terms of its real and imaginary parts, then the function that calculates the principal value Arg is called the <a href="/facts/Atan2/0JDnUYfP">two-argument arctangent function, atan2</a>:

Arg
 ⁡
 (
 x
 +
 i
 y
 )
 =
 atan2
 ⁡
 (
 y
 ,
 
 x
 )
 
 
 {\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)}
 
.
The atan2 function is available in the math libraries of many programming languages, sometimes under a different name, and usually returns a value in the range (−π, π].<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
In some sources the argument is defined as 
 
 
 
 Arg
 ⁡
 (
 x
 +
 i
 y
 )
 =
 arctan
 ⁡
 (
 y
 
 /
 
 x
 )
 ,
 
 
 {\displaystyle \operatorname {Arg} (x+iy)=\arctan(y/x),}
 
 however this is correct only when x > 0, where 
 
 
 
 y
 
 /
 
 x
 
 
 {\displaystyle y/x}
 
 is well-defined and the angle lies between 
 
 
 
 −
 
 
 
 π
 2
 
 
 
 
 
 {\displaystyle -{\tfrac {\pi }{2}}}
 
 and 
 
 
 
 
 
 
 π
 2
 
 
 
 .
 
 
 {\displaystyle {\tfrac {\pi }{2}}.}
 
 Extending this definition to cases where x is not positive is relatively involved. Specifically, one may define the principal value of the argument separately on the half-plane x > 0 and the two quadrants with x < 0, and then patch the definitions together:

Arg
 ⁡
 (
 x
 +
 i
 y
 )
 =
 atan2
 ⁡
 (
 y
 ,
 
 x
 )
 =
 
 
 {
 
 
 
 arctan
 ⁡
 
 (
 
 
 y
 x
 
 
 )
 
 
 
 
 if 
 
 x
 >
 0
 ,
 
 
 
 
 arctan
 ⁡
 
 (
 
 
 y
 x
 
 
 )
 
 +
 π
 
 
 
 if 
 
 x
 <
 0
 
  and 
 
 y
 ≥
 0
 ,
 
 
 
 
 arctan
 ⁡
 
 (
 
 
 y
 x
 
 
 )
 
 −
 π
 
 
 
 if 
 
 x
 <
 0
 
  and 
 
 y
 <
 0
 ,
 
 
 
 
 +
 
 
 π
 2
 
 
 
 
 
 if 
 
 x
 =
 0
 
  and 
 
 y
 >
 0
 ,
 
 
 
 
 −
 
 
 π
 2
 
 
 
 
 
 if 
 
 x
 =
 0
 
  and 
 
 y
 <
 0
 ,
 
 
 
 
 
 undefined
 
 
 
 
 if 
 
 x
 =
 0
 
  and 
 
 y
 =
 0.
 
 
 
 
 
 
 
 
 {\displaystyle \operatorname {Arg} (x+iy)=\operatorname {atan2} (y,\,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\text{if }}x>0,\\[5mu]\arctan \left({\frac {y}{x}}\right)+\pi &{\text{if }}x<0{\text{ and }}y\geq 0,\\[5mu]\arctan \left({\frac {y}{x}}\right)-\pi &{\text{if }}x<0{\text{ and }}y<0,\\[5mu]+{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0,\\[5mu]-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0,\\[5mu]{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}

See <a href="/facts/Atan2/0JDnUYfP">atan2</a> for further detail and alternative implementations.

<h2 id="realizations-of-the-function-in-computer-languages">Realizations of the function in computer languages</h2>
<h3>Wolfram language (Mathematica)</h3>
In Wolfram language, there's Arg[z]:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Arg[x + y I] 
 
 
 
 =
 
 
 {
 
 
 
 
 undefined
 
 
 
 
 if 
 
 
 |
 
 x
 
 |
 
 =
 ∞
 
  and 
 
 
 |
 
 y
 
 |
 
 =
 ∞
 ,
 
 
 
 
 0
 
 
 
 if 
 
 x
 =
 0
 
  and 
 
 y
 =
 0
 ,
 
 
 
 
 0
 
 
 
 if 
 
 x
 =
 ∞
 ,
 
 
 
 
 π
 
 
 
 if 
 
 x
 =
 −
 ∞
 ,
 
 
 
 
 ±
 
 
 π
 2
 
 
 
 
 
 if 
 
 y
 =
 ±
 ∞
 ,
 
 
 
 
 Arg
 ⁡
 (
 x
 +
 y
 i
 )
 
 
 
 otherwise
 
 .
 
 
 
 
 
 
 
 
 {\displaystyle ={\begin{cases}{\text{undefined}}&{\text{if }}|x|=\infty {\text{ and }}|y|=\infty ,\\[5mu]0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]0&{\text{if }}x=\infty ,\\[5mu]\pi &{\text{if }}x=-\infty ,\\[5mu]\pm {\frac {\pi }{2}}&{\text{if }}y=\pm \infty ,\\[5mu]\operatorname {Arg} (x+yi)&{\text{otherwise}}.\end{cases}}}

or using the language's ArcTan:
Arg[x + y I] 
 
 
 
 =
 
 
 {
 
 
 
 0
 
 
 
 if 
 
 x
 =
 0
 
  and 
 
 y
 =
 0
 ,
 
 
 
 
 
 ArcTan[x, y]
 
 
 
 
 otherwise
 
 .
 
 
 
 
 
 
 
 
 {\displaystyle ={\begin{cases}0&{\text{if }}x=0{\text{ and }}y=0,\\[5mu]{\text{ArcTan[x, y]}}&{\text{otherwise}}.\end{cases}}}

ArcTan[x, y] is 
 
 
 
 atan2
 ⁡
 (
 y
 ,
 x
 )
 
 
 {\displaystyle \operatorname {atan2} (y,x)}
 
 extended to work with infinities. ArcTan[0, 0] is Indeterminate (i.e. it's <a href="/facts/Indeterminate_form/5Y3JWKjS">still</a> defined), while ArcTan[Infinity, -Infinity] doesn't return anything (i.e. it's <a href="/facts/Undefined_(mathematics)/Bc1Q3px4">undefined</a>).

<h3>Maple</h3>
<a href="/facts/Maple_(software)/L87oPlmy">Maple</a>'s argument(z) behaves the same as Arg[z] in Wolfram language, except that argument(z) also returns 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 if z is the special floating-point value −0..<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Also, Maple doesn't have 
 
 
 
 atan2
 
 
 {\displaystyle \operatorname {atan2} }
 
.

<h3>MATLAB</h3>
<a href="/facts/MATLAB/qPjLISCk">MATLAB</a>'s angle(z) behaves<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> the same as Arg[z] in Wolfram language, except that it is

 
 
 
 
 
 {
 
 
 
 
 
 
 1
 π
 
 4
 
 
 
 
 
 if 
 
 x
 =
 ∞
 
  and 
 
 y
 =
 ∞
 ,
 
 
 
 
 −
 
 
 
 1
 π
 
 4
 
 
 
 
 
 if 
 
 x
 =
 ∞
 
  and 
 
 y
 =
 −
 ∞
 ,
 
 
 
 
 
 
 
 3
 π
 
 4
 
 
 
 
 
 if 
 
 x
 =
 −
 ∞
 
  and 
 
 y
 =
 ∞
 ,
 
 
 
 
 −
 
 
 
 3
 π
 
 4
 
 
 
 
 
 if 
 
 x
 =
 −
 ∞
 
  and 
 
 y
 =
 −
 ∞
 .
 
 
 
 
 
 
 
 
 {\displaystyle {\begin{cases}{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {1\pi }{4}}&{\text{if }}x=\infty {\text{ and }}y=-\infty ,\\[5mu]{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=\infty ,\\[5mu]-{\frac {3\pi }{4}}&{\text{if }}x=-\infty {\text{ and }}y=-\infty .\end{cases}}}

Unlike in Maple and Wolfram language, MATLAB's atan2(y, x) is equivalent to angle(x + y*1i). That is, atan2(0, 0) is 
 
 
 
 0
 
 
 {\displaystyle 0}
 
.

<h2 id="identities">Identities</h2>
One of the main motivations for defining the principal value Arg is to be able to write complex numbers in modulus-argument form. Hence for any complex number z,

z
        =
        
          |
          z
          |
        
        
          e
          
            i
            Arg
            ⁡
            z
          
        
        .
      
    
    {\displaystyle z=\left|z\right|e^{i\operatorname {Arg} z}.}

This is only really valid if z is non-zero, but can be considered valid for z = 0 if Arg(0) is considered as an <a href="/facts/Indeterminate_form/5Y3JWKjS">indeterminate form</a>—rather than as being undefined.
Some further identities follow. If z1 and z2 are two non-zero complex numbers, then

Arg
                ⁡
                (
                
                  z
                  
                    1
                  
                
                
                  z
                  
                    2
                  
                
                )
              
              
                
                ≡
                Arg
                ⁡
                (
                
                  z
                  
                    1
                  
                
                )
                +
                Arg
                ⁡
                (
                
                  z
                  
                    2
                  
                
                )
                
                  
                  (
                  mod
                  
                  
                    R
                  
                  
                    /
                  
                  2
                  π
                  
                    Z
                  
                  )
                
                ,
              
            
            
              
                Arg
                ⁡
                
                  (
                  
                    
                      
                        z
                        
                          1
                        
                      
                      
                        z
                        
                          2
                        
                      
                    
                  
                  )
                
              
              
                
                ≡
                Arg
                ⁡
                (
                
                  z
                  
                    1
                  
                
                )
                −
                Arg
                ⁡
                (
                
                  z
                  
                    2
                  
                
                )
                
                  
                  (
                  mod
                  
                  
                    R
                  
                  
                    /
                  
                  2
                  π
                  
                    Z
                  
                  )
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\operatorname {Arg} (z_{1}z_{2})&\equiv \operatorname {Arg} (z_{1})+\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }},\\\operatorname {Arg} \left({\frac {z_{1}}{z_{2}}}\right)&\equiv \operatorname {Arg} (z_{1})-\operatorname {Arg} (z_{2}){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.\end{aligned}}}

If z ≠ 0 and n is any integer, then<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

Arg
        ⁡
        
          (
          
            z
            
              n
            
          
          )
        
        ≡
        n
        Arg
        ⁡
        (
        z
        )
        
          
          (
          mod
          
          
            R
          
          
            /
          
          2
          π
          
            Z
          
          )
        
        .
      
    
    {\displaystyle \operatorname {Arg} \left(z^{n}\right)\equiv n\operatorname {Arg} (z){\pmod {\mathbb {R} /2\pi \mathbb {Z} }}.}

<h3>Example</h3>

Arg
        ⁡
        
          
            (
          
        
        
          
            
              −
              1
              −
              i
            
            i
          
        
        
          
            )
          
        
        =
        Arg
        ⁡
        (
        −
        1
        −
        i
        )
        −
        Arg
        ⁡
        (
        i
        )
        =
        −
        
          
            
              3
              π
            
            4
          
        
        −
        
          
            π
            2
          
        
        =
        −
        
          
            
              5
              π
            
            4
          
        
      
    
    {\displaystyle \operatorname {Arg} {\biggl (}{\frac {-1-i}{i}}{\biggr )}=\operatorname {Arg} (-1-i)-\operatorname {Arg} (i)=-{\frac {3\pi }{4}}-{\frac {\pi }{2}}=-{\frac {5\pi }{4}}}

<h3>Using the complex logarithm</h3>
From 
 
 
 
 z
 =
 
 |
 
 z
 
 |
 
 
 e
 
 i
 Arg
 ⁡
 (
 z
 )
 
 
 
 
 {\displaystyle z=|z|e^{i\operatorname {Arg} (z)}}
 
, we get 
 
 
 
 i
 Arg
 ⁡
 (
 z
 )
 =
 ln
 ⁡
 
 
 z
 
 
 |
 
 z
 
 |
 
 
 
 
 
 
 {\displaystyle i\operatorname {Arg} (z)=\ln {\frac {z}{|z|}}}
 
, alternatively 
 
 
 
 Arg
 ⁡
 (
 z
 )
 =
 Im
 ⁡
 (
 ln
 ⁡
 
 
 z
 
 
 |
 
 z
 
 |
 
 
 
 
 )
 =
 Im
 ⁡
 (
 ln
 ⁡
 z
 )
 
 
 {\displaystyle \operatorname {Arg} (z)=\operatorname {Im} (\ln {\frac {z}{|z|}})=\operatorname {Im} (\ln z)}
 
. As we are taking the imaginary part, any normalisation by a real scalar will not affect the result. This is useful when one has the <a href="/facts/Complex_logarithm/X2U0YQuG">complex logarithm</a> available.

<h2 id="extended-argument">Extended argument</h2>
The extended argument of a number z (denoted as 
 
 
 
 
 
 arg
 ¯
 
 
 (
 z
 )
 
 
 {\displaystyle {\overline {\arg }}(z)}
 
) is the set of all real numbers congruent to 
 
 
 
 arg
 ⁡
 (
 z
 )
 
 
 {\displaystyle \arg(z)}
 
 modulo 2
 
 
 
 π
 
 
 {\displaystyle \pi }
 
.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
 
 
 
 
 
 arg
 ¯
 
 
 (
 z
 )
 =
 arg
 ⁡
 (
 z
 )
 +
 2
 k
 π
 ,
 ∀
 k
 ∈
 
 Z
 
 
 
 {\displaystyle {\overline {\arg }}(z)=\arg(z)+2k\pi ,\forall k\in \mathbb {Z} }

<h2 id="bibliography">Bibliography</h2>

<ul><li>Ahlfors, Lars (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3rd ed.). New York;London: McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-000657-1.</li>
<li>Ponnuswamy, S. (2005). Foundations of Complex Analysis (2nd ed.). New Delhi;Mumbai: Narosa. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-81-7319-629-4.</li>
<li>Beardon, Alan (1979). Complex Analysis: The Argument Principle in Analysis and Topology. Chichester: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-99671-8.</li>
<li>Borowski, Ephraim; Borwein, Jonathan (2002) [1st ed. 1989 as Dictionary of Mathematics]. Mathematics. Collins Dictionary (2nd ed.). Glasgow: <a href="/facts/HarperCollins/HHLz5lk5">HarperCollins</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-00-710295-X.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php/Argument">Argument</a> at <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31. <a href="https://mathworld.wolfram.com/ComplexArgument.html" target="_blank">https://mathworld.wolfram.com/ComplexArgument.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Modulus and Argument". mas-coursebuild.ncl.ac.uk. Newcastle University. Retrieved 2025-01-05. <a href="https://mas-coursebuild.ncl.ac.uk/lti/content/ASK/default/core_mathematics/pure_maths/algebra/complex_numbers/modulus_and_argument/index.html" target="_blank">https://mas-coursebuild.ncl.ac.uk/lti/content/ASK/default/core_mathematics/pure_maths/algebra/complex_numbers/modulus_and_argument/index.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Modulus and Argument of a Complex Number". Byju's. Retrieved 18 January 2025. <a href="https://byjus.com/jee/complex-numbers/#modulus-and-argument-of-a-complex-number" target="_blank">https://byjus.com/jee/complex-numbers/#modulus-and-argument-of-a-complex-number</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Dictionary of Mathematics (2002). phase. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31. <a href="https://mathworld.wolfram.com/ComplexArgument.html" target="_blank">https://mathworld.wolfram.com/ComplexArgument.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31. <a href="https://mathworld.wolfram.com/ComplexArgument.html" target="_blank">https://mathworld.wolfram.com/ComplexArgument.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Arg". Wolfram Language Documentation. Retrieved 2024-08-30. <a href="https://reference.wolfram.com/language/ref/Arg.html" target="_blank">https://reference.wolfram.com/language/ref/Arg.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"Argument - Maple Help". <a href="https://www.maplesoft.com/support/help/Maple/view.aspx?path=argument" target="_blank">https://www.maplesoft.com/support/help/Maple/view.aspx?path=argument</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">"Phase angle - MATLAB angle". <a href="https://www.mathworks.com/help/matlab/ref/angle.html" target="_blank">https://www.mathworks.com/help/matlab/ref/angle.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"Four-quadrant inverse tangent - MATLAB atan2". <a href="https://www.mathworks.com/help/matlab/ref/atan2.html" target="_blank">https://www.mathworks.com/help/matlab/ref/atan2.html</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Weisstein, Eric W. "Complex Argument". mathworld.wolfram.com. Retrieved 2020-08-31. <a href="https://mathworld.wolfram.com/ComplexArgument.html" target="_blank">https://mathworld.wolfram.com/ComplexArgument.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Algebraic Structure of Complex Numbers". www.cut-the-knot.org. Retrieved 2021-08-29. <a href="https://www.cut-the-knot.org/arithmetic/algebra/ComplexNumbers.shtml" target="_blank">https://www.cut-the-knot.org/arithmetic/algebra/ComplexNumbers.shtml</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Argument (complex analysis) open-in-new

Argument (complex analysis)