Imaginary unit

Imaginary unit

The imaginary unit or unit imaginary number (i) is a solution to the <a href="/en/Quadratic_equation" title="Quadratic equation" class="wl">quadratic equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1=0}">
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 <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0}">. Although there is no <a href="/en/Real_number" title="Real number" class="wl">real number</a> with this property, i can be used to extend the real numbers to what are called <a href="/en/Complex_number" title="Complex number" class="wl">complex numbers</a>, using <a href="/en/Addition" title="Addition" class="wl">addition</a> and <a href="/en/Multiplication" title="Multiplication" class="wl">multiplication</a>. A simple example of the use of i in a complex number is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+3i}">
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 <annotation encoding="application/x-tex">{\displaystyle 2+3i}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8f8108e96052f0db5c8d98246a22d1764b1e3c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:5.968ex; height:2.343ex;" alt="{\displaystyle 2+3i}">.

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Principal square root of −1</div>
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<figure class="mw-default-size mw-halign-right wiki-thumb"><a href="/en/File:ImaginaryUnit5.svg" class="mw-file-description image media-thumb" data-hash="File:ImaginaryUnit5.svg"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/220px-ImaginaryUnit5.svg.png" decoding="async" data-file-width="480" data-file-height="480" data-file-type="drawing" height="220" width="220" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/330px-ImaginaryUnit5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/ImaginaryUnit5.svg/440px-ImaginaryUnit5.svg.png 2x" alt="ImaginaryUnit5.svg"></a><figcaption>i in the <a href="/en/Complex_plane" title="Complex plane" class="wl">complex</a> or <a href="/en/Cartesian_plane" title="Cartesian plane" class="mw-redirect wl">Cartesian plane</a>. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis.</figcaption></figure>

<a href="/en/Imaginary_number" title="Imaginary number" class="wl">Imaginary numbers</a> are an important mathematical concept; they extend the real number system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\mathbb {R} "> to the complex number system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">
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 <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\mathbb {C} ">, in which at least one <a href="/en/Root_of_a_function" title="Root of a function" class="mw-redirect wl">root</a> for every nonconstant <a href="/en/Polynomial" title="Polynomial" class="wl">polynomial</a> exists (see <a href="/en/Algebraic_closure" title="Algebraic closure" class="wl">Algebraic closure</a> and <a href="/en/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra" class="wl">Fundamental theorem of algebra</a>). Here, the term "imaginary" is used because there is no <a href="/en/Real_number" title="Real number" class="wl">real number</a> having a negative <a href="/en/Square_(algebra)" title="Square (algebra)" class="wl">square</a>.

There are two complex square roots of −1: i and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i}">
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="-i">, just as there are two complex <a href="/en/Square_root" title="Square root" class="wl">square roots</a> of every real number other than <a href="/en/Zero" title="Zero" class="mw-redirect wl">zero</a> (which has one <a href="/en/Multiple_root" title="Multiple root" class="mw-redirect wl">double square root</a>).

In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in <a href="/en/Electrical_engineering" title="Electrical engineering" class="wl">electrical engineering</a> and <a href="/en/Control_systems_engineering" title="Control systems engineering" class="mw-redirect wl">control systems engineering</a>, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote <a href="/en/Electric_current" title="Electric current" class="wl">electric current</a>.<a href="#cite_note-1" style="counter-reset: mw-Ref 1;">[1]</a>

The imaginary unit or unit imaginary number is a solution to the quadratic equation 
  
    
      
        
          x
          
            2
          
        
        +
        1
        =
        0
      
    
    {\displaystyle x^{2}+1=0}
  
. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 
  
    
      
        2
        +
        3
        i
      
    
    {\displaystyle 2+3i}
  
. Imaginary numbers are an important mathematical concept; they extend the real number system 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 to the complex number system 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
, in which at least one root for every nonconstant polynomial exists . Here, the term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of −1: i and 
  
    
      
        −
        i
      
    
    {\displaystyle -i}
  
, just as there are two complex square roots of every real number other than zero . 