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Reference.org
Imaginary number
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<table><tbody><tr><th>The powers of i are cyclic:</th></tr><tr><td> ⋮ {\displaystyle \ \vdots } </td></tr><tr><td> i − 2 = − 1 i {\displaystyle \ i^{-2}=-1{\phantom {i}}} </td></tr><tr><td> i − 1 = − i 1 {\displaystyle \ i^{-1}=-i{\phantom {1}}} </td></tr><tr><td> i 0 = − 1 i {\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}} </td></tr><tr><td> i 1 = − i 1 {\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}} </td></tr><tr><td> i 2 = − 1 i {\displaystyle \ \ i^{2}\ =-1{\phantom {i}}} </td></tr><tr><td> i 3 = − i 1 {\displaystyle \ \ i^{3}\ =-i{\phantom {1}}} </td></tr><tr><td> i 4 = − 1 i {\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}} </td></tr><tr><td> i 5 = − i 1 {\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}} </td></tr><tr><td> ⋮ {\displaystyle \ \vdots } </td></tr><tr><td> i {\displaystyle i} is a 4th <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a></td></tr></tbody></table> <p>An imaginary number is the product of a <a href="/facts/Real_number/R02gw5Pb">real number</a> and the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> i, which is defined by its property <i>i</i>2 = −1. The <a href="/facts/Square_(algebra)/Jzpq0R0z">square</a> of an imaginary number bi is −<i>b</i>2. For example, 5<i>i</i> is an imaginary number, and its square is −25. The number <a href="/facts/0/4NBRtKHc">zero</a> is considered to be both real and imaginary. </p><p>Originally coined in the 17th century by <a href="/facts/Ren%25C3%25A9_Descartes/ohqj6DMf">René Descartes</a> as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> (in the 18th century) and <a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a> and <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a> (in the early 19th century). </p><p>An imaginary number <i>bi</i> can be added to a real number a to form a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> of the form <i>a</i> + <i>bi</i>, where the real numbers a and b are called, respectively, the <i>real part</i> and the <i>imaginary part</i> of the complex number. </p>