Editing Imaginary unit (section)

{{Short description|Principal square root of minus 1}}
{{redirect|i (number)|internet numbers|i-number}}
{{use dmy dates|date=August 2020}}

[[File:ImaginaryUnit5.svg|thumb|right|The imaginary unit {{mvar|i}} in the [[complex plane]]: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.]]

The '''imaginary unit''' or '''unit imaginary number''' ('''{{mvar|i}}''') is a [[mathematical constant]] that is a solution to the [[quadratic equation]] {{math|1=''x''{{isup|2}} + 1 = 0.}} Although there is no [[real number]] with this property, {{mvar|i}} can be used to extend the real numbers to what are called [[complex number]]s, using [[addition]] and [[multiplication]]. A simple example of the use of {{mvar|i}} in a complex number is {{math|2 + 3''i''.}}

[[Imaginary number]]s are an important mathematical concept; they extend the real number system <math>\mathbb{R}</math> to the complex number system <math>\mathbb{C},</math> in which at least one [[Root of a function|root]] for every nonconstant [[polynomial]] exists (see [[Algebraic closure]] and [[Fundamental theorem of algebra]]). Here, the term ''imaginary'' is used because there is no [[real number]] having a negative [[square (algebra)|square]].

There are two complex square roots of {{math|−1:}} {{mvar|i}} and {{math|−''i''}}, just as there are two complex [[square root]]s of every real number other than [[zero]] (which has one [[multiple root|double square root]]).

In contexts in which use of the letter {{mvar|i}} is ambiguous or problematic, the letter {{mvar|j}} is sometimes used instead. For example, in [[electrical engineering]] and [[control systems engineering]], the imaginary unit is normally denoted by {{mvar|j}} instead of {{mvar|i}}, because {{mvar|i}} is commonly used to denote [[electric current]].<ref>{{cite book |last=Stubbings |first=George Wilfred |year=1945 |title=Elementary vectors for electrical engineers |place=London |publisher=I. Pitman |page=69 |url=https://archive.org/details/elementaryvector00stub/page/69/ |url-access=limited }} {{pb}} {{cite book |last=Boas |first=Mary L. |title=Mathematical Methods in the Physical Sciences |year=2006 |edition=3rd |publisher=Wiley |location=New York [u.a.] |isbn=0-471-19826-9 |page=49}}</ref>