Editing Imaginary unit

{{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>

== Terminology ==
{{Further|Complex number#History}}
Square roots of negative numbers are called ''imaginary'' because in [[History of mathematics#Renaissance|early-modern mathematics]], only what are now called [[real numbers]], obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even [[negative numbers]] were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical. The name ''imaginary'' is generally credited to [[René Descartes]], and [[Isaac Newton]] used the term as early as 1670.<ref>{{cite journal|title=The New Language of Mathematics |last=Silver |first=Daniel S.|journal=[[American Scientist]] |volume=105 |number=6 |date=November–December 2017 |pages=364–371 |doi=10.1511/2017.105.6.364 |url=https://www.americanscientist.org/article/the-new-language-of-mathematics}}</ref><ref>{{cite OED|imaginary number}}</ref> The {{mvar|i}} notation was introduced by [[Leonhard Euler]].<ref name=Boyer>{{cite book|title = A History of Mathematics|last1 = Boyer|first1 = Carl B.|author-link=Carl Benjamin Boyer|last2 = Merzbach |first2=Uta C.|author2-link = Uta Merzbach|publisher = [[John Wiley & Sons]]|isbn = 978-0-471-54397-8|pages = [https://archive.org/details/historyofmathema00boye/page/439 439–445]|year = 1991|url = https://archive.org/details/historyofmathema00boye/page/439}}</ref>

A ''unit'' is an undivided whole, and ''unity'' or the ''unit number'' is the number [[1|one]] ({{math|1}}).

==Definition==
{| class="wikitable" style="float: right; margin-left: 1em; text-align: center;"
! The powers of {{mvar|i}}<br/> are cyclic:
|-
|<math>\ \vdots</math>
|-
|<math>\ i^{-4} = \phantom-1\phantom{i}</math>
|-
|<math>\ i^{-3} = \phantom-i\phantom1</math>
|-
|<math>\ i^{-2} = -1\phantom{i}</math>
|-
|<math>\ i^{-1} = -i\phantom1</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{0}\ = \phantom-1\phantom{i}</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{1}\  = \phantom-i\phantom1</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{2}\  = -1\phantom{i}</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{3}\  = -i\phantom1</math>
|-
|<math>\ \ i^{4}\  = \phantom-1\phantom{i}</math>
|-
|<math>\ \ i^{5}\  = \phantom-i\phantom1</math>
|-
|<math>\ \ i^{6}\  = -1\phantom{i}</math>
|-
|<math>\ \ i^{7}\  = -i\phantom1</math>
|-
|<math>\ \vdots</math>
|}

The imaginary unit {{mvar|i}} is defined solely by the property that its square is −1:
<math display=block>i^2 = -1.</math>

With {{mvar|i}} defined this way, it follows directly from [[algebra]] that {{mvar|i}} and {{math|−''i''}} are both square roots of −1.

Although the construction is called ''imaginary'', and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating {{mvar|i}} as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of {{math|''i''{{isup|2}}}} with {{math|−1}}). Higher integral powers of {{mvar|i}} are thus
<math display=block>\begin{alignat}{3}
i^3 &= i^2 i &&= (-1) i &&= -i, \\[3mu]
i^4 &= i^3 i &&= \;\!(-i) i &&= \ \,1, \\[3mu]
i^5 &= i^4 i &&= \ \, (1) i &&= \ \ i,
\end{alignat}</math>
and so on, cycling through the four values {{math|1}}, {{mvar|i}}, {{math|−1}}, and {{math|−''i''}}. As with any non-zero real number, {{math|1=''i''{{isup|0}} = 1.}}

As a complex number, {{mvar|i}} can be represented in [[Rectangular coordinate system|rectangular form]] as {{math|0 + 1''i''}}, with a zero real component and a unit imaginary component. In [[polar form]], {{mvar|i}} can be represented as {{math|1 × ''e''{{isup|''πi'' /2}}}} (or just {{math|''e''{{isup|''πi'' /2}}}}), with an [[absolute value]] (or magnitude) of 1 and an [[argument (complex analysis)|argument]] (or angle) of <math>\tfrac\pi2</math> [[radian]]s. (Adding any integer multiple of {{math|2''π''}} to this angle works as well.)  In the [[complex plane]], which is a special interpretation of a [[Cartesian plane]], {{mvar|i}} is the point located one unit from the origin along the [[imaginary axis]] (which is [[perpendicular]] to the [[real axis]]).

==={{math|''i''}} vs. {{math|−''i''}}===
{{anchor|i and -i}}
Being a [[quadratic polynomial]] with no [[multiple root]], the defining equation {{math|1=''x''{{isup|2}} = −1}} has {{em|two}} distinct solutions, which are equally valid and which happen to be [[additive inverse|additive]] and [[multiplicative inverse]]s of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled {{math|+''i''}} (or simply {{mvar|i}}) and the other is labelled {{math|−''i''}}, though it is inherently ambiguous which is which.

The only differences between {{math|+''i''}} and {{math|−''i''}} arise from this labelling. For example, by convention {{math|+''i''}} is said to have an [[Argument (complex analysis)|argument]] of <math>+\tfrac\pi2</math> and {{math|−''i''}} is said to have an argument of <math>-\tfrac\pi2,</math> related to the convention of labelling orientations in the [[Cartesian plane]] relative to the positive {{mvar|x}}-axis with positive angles turning [[anticlockwise]] in the direction of the positive {{mvar|y}}-axis. Also, despite the signs written with them, neither {{math|+''i''}} nor {{math|−''i''}} is inherently positive or negative in the sense that real numbers are.<ref>{{cite book |first1=Apostolos K. |last1=Doxiadēs |first2=Barry |last2=Mazur |year=2012 |title=Circles Disturbed: The interplay of mathematics and narrative |page= [https://books.google.com/books?id=X9Uoug4lNWkC&pg=PA225 225] |edition=illustrated |publisher=Princeton University Press |isbn=978-0-691-14904-2 |via=Google Books |url=https://books.google.com/books?id=X9Uoug4lNWkC}}</ref>

A more formal expression of this indistinguishability of {{math|+''i''}} and {{math|−''i''}} is that, although the complex [[field (algebra)|field]] is [[unique (mathematics)|unique]] (as an extension of the real numbers) [[up to]] [[isomorphism]], it is {{em|not}} unique up to a {{em|unique}} isomorphism. That is, there are two [[automorphism|field automorphisms]] of the complex numbers <math>\C</math> that keep each real number fixed, namely the identity and [[complex conjugation]]. For more on this general phenomenon, see [[Galois group]].

===Matrices===
Using the concepts of [[matrix (mathematics)|matrices]]  and [[matrix multiplication]], complex numbers can be represented in linear algebra. The real unit {{math|1}} and imaginary unit {{mvar|i}} can be represented by any pair of matrices {{mvar|I}} and {{mvar|J}} satisfying {{math|1=''I''{{isup|2}} = ''I'',}} {{math|1=''IJ'' = ''JI'' = ''J'',}} and {{math|1=''J''{{isup|2}} = −''I''.}} Then a complex number {{math|''a'' + ''bi''}} can be represented by the matrix {{math|''aI'' + ''bJ'',}} and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.

The most common choice is to represent {{math|1}} and {{mvar|i}} by the {{math|2&hairsp;×&hairsp;2}} [[identity matrix]] {{mvar|I}} and the matrix {{mvar|J}},

<math display=block>
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.</math>

Then an arbitrary complex number {{math|''a'' + ''bi''}} can be represented by:

<math display=block>aI + bJ = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.</math>

More generally, any real-valued {{math|2&hairsp;×&hairsp;2}} matrix with a [[trace (linear algebra)|trace]] of zero and a [[determinant]] of one squares to {{math|−''I''}}, so could be chosen for {{mvar|J}}. Larger matrices could also be used; for example, {{math|1}} could be represented by the {{math|4&hairsp;×&hairsp;4}} identity matrix and {{mvar|i}} could be represented by any of the [[Dirac matrices]] for spatial dimensions.

===Root of {{math|''x''<sup>2</sup> + 1}}===
[[Polynomial]]s (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose [[coefficient]]s are real numbers form a [[ring (mathematics)|ring]], denoted <math>\R[x],</math> an algebraic structure with addition and multiplication and sharing many properties with the ring of [[integer]]s.

The polynomial <math>x^2 + 1</math> has no real-number [[root of a polynomial|roots]], but the set of all real-coefficient polynomials divisible by <math>x^2 + 1</math> forms an [[ideal (ring theory)|ideal]], and so there is a [[Polynomial ring#Quotient ring|quotient ring]] <math>\reals[x] / \langle x^2 + 1\rangle.</math> This quotient ring is [[isomorphism|isomorphic]] to the complex numbers, and the variable <math>x</math> expresses the imaginary unit.

=== Graphic representation ===
{{main|Complex plane}}
The complex numbers can be represented graphically by drawing the real [[number line]] as the horizontal axis and the imaginary numbers as the vertical axis of a [[Cartesian coordinate system|Cartesian plane]] called the ''[[complex plane]]''. In this representation, the numbers {{math|1}} and {{mvar|i}} are at the same distance from {{math|0}}, with a right angle between them. Addition by a complex number corresponds to [[translation (geometry)|translation]] in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every [[similarity (geometry)|similarity]] transformation of the plane can be represented by a complex-linear function <math>z \mapsto az + b.</math>

=== Geometric algebra ===

In the [[geometric algebra]] of the [[Euclidean plane]], the geometric product or quotient of two arbitrary [[Euclidean vector|vectors]] is a sum of a scalar (real number) part and a [[bivector]] part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.

The quotient of a vector with itself is the scalar {{math|1=1 = ''u''/''u''}}, and when multiplied by any vector leaves it unchanged (the [[Identity function|identity transformation]]). The quotient of any two perpendicular vectors of the same magnitude, {{math|1=''J'' = ''u''/''v''}}, which when multiplied rotates the divisor a quarter turn into the dividend, {{math|1=''Jv'' = ''u''}}, is a unit bivector which squares to {{math|&minus;1}}, and can thus be taken as a representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is [[isomorphic]] to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.<ref>The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by [[Hermann Grassmann]] in the foreword to his ''Ausdehnungslehre'' of 1844; later [[William Kingdon Clifford|William Clifford]] realized that this ratio could be interpreted as a bivector. {{pb}} {{cite book |last=Hestenes |first=David |author-link=David Hestenes |year=1996 |chapter=Grassmann’s Vision |editor-last=Schubring |editor-first=G. |title=Hermann Günther Graßmann (1809–1877) |series=Boston Studies in the Philosophy of Science |volume=187 |pages=243–254 |publisher=Springer |doi=10.1007/978-94-015-8753-2_20 |isbn=978-90-481-4758-8 |chapter-url=https://davidhestenes.net/geocalc/pdf/GrassmannsVision.pdf }}</ref>

More generally, in the geometric algebra of any higher-dimensional [[Euclidean space]], a unit bivector of any arbitrary planar orientation squares to {{math|−1}}, so can be taken to represent the imaginary unit {{mvar|i}}.

==Proper use==
The imaginary unit was historically written <math display=inline>\sqrt{-1},</math> and still is in some modern works. However, great care needs to be taken when manipulating formulas involving [[Nth root|radicals]]. The radical sign notation <math display=inline>\sqrt{x}</math> is reserved either for the principal square root function, which is defined for ''only'' real {{math|''x'' ≥ 0,}} or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results:<ref>{{cite book |first=Bryan |last=Bunch |year=2012 |title=Mathematical Fallacies and Paradoxes |edition=illustrated |publisher=Courier Corporation |page=[https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA31 31]-34 |isbn=978-0-486-13793-3 |via=Google Books |url=https://books.google.com/books?id=jUTCAgAAQBAJ}}</ref>
<math display=block>-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} \mathrel{\stackrel{\mathrm{fallacy}}{=}} {\textstyle \sqrt{(-1) \cdot (-1)}} = \sqrt{1} = 1 \qquad \text{(incorrect).}</math>

Generally, the calculation rules
<math display=inline>\sqrt{x\vphantom{ty}} \cdot\! \sqrt{y\vphantom{ty}} = \sqrt{x \cdot y\vphantom{ty}}</math>
and
<math display=inline>\sqrt{x\vphantom{ty}}\big/\!\sqrt{y\vphantom{ty}} = \sqrt{x/y}</math>
are guaranteed to be valid only for real, positive values of {{mvar|x}} and {{mvar|y}}.<ref>{{cite book |first=Arthur |last=Kramer |publisher=Cengage Learning |year=2012 |edition=4th |title=Math for Electricity & Electronics |isbn=978-1-133-70753-0 |page=[https://books.google.com/books?id=gdAJAAAAQBAJ&pg=PA81 81] |via=Google Books |url=https://books.google.com/books?id=gdAJAAAAQBAJ}}</ref><ref>{{cite book |last1=Picciotto |first1=Henri |last2=Wah |first2=Anita |year=1994 |title=Algebra: Themes, tools, concepts |edition=Teachers' |publisher=Henri Picciotto |isbn=978-1-56107-252-1 |page=[https://books.google.com/books?id=_cOhDl3J3ZMC&pg=PA424 424] |via=Google Books |url=https://books.google.com/books?id=_cOhDl3J3ZMC}}</ref><ref>{{cite book |first=Paul J. |last=Nahin |year=2010 |title=An Imaginary Tale: The story of "{{mvar|i}}" [the square root of minus one] |publisher=Princeton University Press |isbn=978-1-4008-3029-9 |page=[https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 12] |via=Google Books |url=https://books.google.com/books?id=PflwJdPhBlEC}}</ref>

When {{mvar|x}} or {{mvar|y}} is real but negative, these problems can be avoided by writing and manipulating expressions like <math display=inline>i \sqrt{7}</math>, rather than <math display=inline>\sqrt{-7}</math>. For a more thorough discussion, see the articles [[Square root]] and [[Branch point]].

== Properties ==

As a complex number, the imaginary unit follows all of the rules of [[Complex number#Relations and operations|complex arithmetic]].

=== Imaginary integers and imaginary numbers ===

When the imaginary unit is repeatedly added or subtracted, the result is some [[integer]] times the imaginary unit, an ''imaginary integer''; any such numbers can be added and the result is also an imaginary integer:

<math display=block>ai + bi = (a + b)i.</math>

Thus, the imaginary unit is the generator of a [[group (mathematics)|group]] under addition, specifically an infinite [[cyclic group]].

The imaginary unit can also be multiplied by any arbitrary [[real number]] to form an [[imaginary number]]. These numbers can be pictured on a [[number line]], the ''imaginary axis'', which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.

=== Gaussian integers ===

Integer sums of the real unit {{math|1}} and the imaginary unit {{mvar|i}} form a [[square lattice]] in the complex plane called the [[Gaussian integers]]. The sum, difference, or product of Gaussian integers is also a Gaussian integer:

<math display=block>\begin{align}
(a + bi) + (c + di) &= (a + c) + (b + d)i, \\[5mu]
(a + bi)(c + di) &= (ac - bd) + (ad + bc)i.
\end{align}</math>

=== Quarter-turn rotation ===

When multiplied by the imaginary unit {{mvar|i}}, any arbitrary complex number in the complex plane is rotated by a quarter turn {{nobr|(<math>\tfrac12\pi</math> radians}} or {{math|90°}}) [[anticlockwise]]. When multiplied by {{math|−''i''}}, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:

<math display=block>i \, re^{\varphi i} = re^{(\varphi + \pi/2)i}, \quad
        -i \, re^{\varphi i} = re^{(\varphi - \pi/2)i}.</math>

In rectangular form,

<math display=block> i(a + bi) = -b + ai, \quad -i(a + bi) = b - ai.</math>

=== Integer powers ===
The powers of {{mvar|i}} repeat in a cycle expressible with the following pattern, where {{mvar|n}} is any integer:

<math display=block>
i^{4n} = 1, \quad
i^{4n+1} = i, \quad
i^{4n+2} = -1, \quad
i^{4n+3} = -i.</math>

Thus, under multiplication, {{mvar|i}} is a generator of a [[cyclic group]] of order 4, a discrete subgroup of the continuous [[circle group]] of the unit complex numbers under multiplication.

Written as a special case of [[Euler's formula]] for an integer {{mvar|n}},

<math display=block>
i^n = {\exp}\bigl(\tfrac12\pi i\bigr)^n
    = {\exp}\bigl(\tfrac12 n \pi i\bigr)
    = {\cos}\bigl(\tfrac12 n\pi \bigr) + {i \sin}\bigl(\tfrac12 n\pi \bigr).
</math>

With a careful choice of [[branch cut]]s and [[principal value]]s, this last equation can also apply to arbitrary complex values of {{mvar|n}}, including cases like {{math|1=''n'' = ''i''}}.{{cn|date=March 2024}}

=== Roots ===
[[File:Imaginary2Root.svg|thumb|right|200px|The two square roots of {{mvar|i}} in the complex plane]]
Just like all nonzero complex numbers, <math display=inline>i = e^{\pi i/ 2}</math> has two distinct [[square root]]s which are [[additive inverse]]s. In polar form, they are
<math display=block>\begin{alignat}{3}
\sqrt{i} &= {\exp}\bigl(\tfrac12{\pi i}\bigr)^{1/2} &&{}= {\exp}\bigl(\tfrac14\pi i\bigr), \\
-\sqrt{i} &= {\exp}\bigl(\tfrac14{\pi i}-\pi i\bigr) &&{}= {\exp}\bigl({-\tfrac34\pi i}\bigr).
\end{alignat}</math>

In rectangular form, they are{{efn|To find such a number, one can solve the equation {{math|1=(''x'' + ''iy''){{sup|2}} = ''i''}} where {{mvar|x}} and {{mvar|y}} are real parameters to be determined, or equivalently {{math|1=''x''{{isup|2}} + 2''ixy'' − ''y''{{isup|2}} = ''i''.}} Because the real and imaginary parts are always separate, we regroup the terms, {{math|1=''x''{{isup|2}} − ''y''{{isup|2}} + 2''ixy'' = 0 + ''i''.}} By [[equating coefficients]], separating the real part and imaginary part, we have a system of two equations:

<math display=block>\begin{align}
x^{2} - y^{2} &= 0 \\[3mu]
2xy &= 1.
\end{align}</math>

Substituting <math display=inline>y=\tfrac12x^{-1}</math> into the first equation, we get <math display=inline> x^{2} - \tfrac14x^{-2} = 0</math> <math display=inline>\implies 4x^4 = 1.</math> Because {{mvar|x}} is a real number, this equation has two real solutions for {{mvar|x}}<math display=block>x=\tfrac{1}{\sqrt{2}}</math> and <math>x=-\tfrac{1}{\sqrt{2}}</math>. Substituting either of these results into the equation {{math|1=2''xy'' = 1}} in turn, we will get the corresponding result for {{mvar|y}}. Thus, the square roots of {{mvar|i}} are the numbers <math>\tfrac{1}{\sqrt{2}} + \tfrac{1}{\sqrt{2}}i</math> and <math>-\tfrac{1}{\sqrt{2}}-\tfrac{1}{\sqrt{2}}i</math>.<ref>{{cite web |website=University of Toronto Mathematics Network |title=What is the square root of {{mvar|i}}&nbsp;? |access-date=26 March 2007 |url=http://www.math.utoronto.ca/mathnet/questionCorner/rootofi.html}}</ref>}}

<math display=block>\begin{alignat}{3}
\sqrt{i} &= \frac{1 + i}{ \sqrt{2}} &&{}= \phantom{-}\tfrac{\sqrt{2}}{2} + \tfrac{\sqrt{2}}{2}i, \\[5mu]
-\sqrt{i} &= - \frac{1 + i}{ \sqrt{2}} &&{}= - \tfrac{\sqrt{2}}{2} - \tfrac{\sqrt{2}}{2}i.
\end{alignat}</math>

Squaring either expression yields
<math display=block>
\left( \pm \frac{1 + i}{\sqrt2} \right)^2
= \frac{1 + 2i - 1}{2} = \frac{2i}{2} = i.
</math>

[[File:Imaginary3Root.svg|thumb|right|200px|The three cube roots of {{mvar|i}} in the complex plane]]
The three [[cube root]]s of {{mvar|i}} are<ref>{{Cite book |last1=Zill |first1=Dennis G. |url=https://www.worldcat.org/oclc/50495529 |title=A first course in complex analysis with applications |last2=Shanahan |first2=Patrick D. |year=2003 |publisher=Jones and Bartlett |isbn=0-7637-1437-2 |location=Boston |pages=24–25 |oclc=50495529}}</ref>

<math display=block>
\sqrt[3]i = {\exp}\bigl(\tfrac16 \pi i\bigr) = \tfrac{\sqrt{3}}{2} + \tfrac12i, \quad
            {\exp}\bigl(\tfrac56 \pi i\bigr) = -\tfrac{\sqrt{3}}{2} + \tfrac12i, \quad
            {\exp}\bigl({-\tfrac12 \pi i}\bigr) = -i.
</math>

For a general positive integer {{mvar|n}}, the [[nth root|{{mvar|n}}-th roots]] of {{mvar|i}} are, for {{math|1=''k'' = 0, 1, ..., ''n'' − 1,}}
<math display=block>
\exp \left(2 \pi i \frac{k+\frac14}{n} \right)
= \cos \left(\frac{4k+1}{2n}\pi \right) +
  i \sin \left(\frac{4k+1}{2n}\pi \right).
</math>
The value associated with {{math|1=''k'' = 0}} is the [[principal value|principal]] {{mvar|n}}-th root of {{mvar|i}}.  The set of roots equals the corresponding set of [[root of unity|roots of unity]] rotated by the principal {{mvar|n}}-th root of {{mvar|i}}. These are the vertices of a [[regular polygon]] inscribed within the complex [[unit circle]].

=== Exponential and logarithm ===

The [[complex exponential]] function relates complex addition in the domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with {{math|1}} representing multiplication by {{mvar|e}}, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with {{mvar|i}} representing a rotation by {{math|1}} radian. The complex exponential is thus a periodic function in the imaginary direction, with period {{math|2''πi''}} and image {{math|1}} at points {{math|2''kπi''}} for all integers {{mvar|k}}, a real multiple of the lattice of imaginary integers.

The complex exponential can be broken into [[even and odd functions|even and odd]] components, the [[hyperbolic functions]] {{math|cosh}} and {{math|sinh}} or the [[trigonometric functions]] {{math|cos}} and {{math|sin}}:

<math display=block>\exp z = \cosh z + \sinh z = \cos(-iz) + i\sin(-iz)</math>

[[Euler's formula]] decomposes the exponential of an imaginary number representing a rotation:

<math display="block">\exp i\varphi = \cos \varphi + i\sin \varphi.</math>

This fact can be used to demonstrate, among other things, the apparently counterintuitive result that <math>i^i</math> is a real number.<ref>{{Cite web |title=i to the i is a Real Number – Math Fun Facts |url=https://math.hmc.edu/funfacts/i-to-the-i-is-a-real-number/ |access-date=2024-08-22 |website=math.hmc.edu |language=en-US}}</ref>

The quotient {{math|1=coth ''z'' = cosh ''z'' / sinh ''z'',}} with appropriate scaling, can be represented as an infinite [[partial fraction decomposition]] as the sum of [[reciprocal function]]s translated by imaginary integers:<ref>Euler expressed the partial fraction decomposition of the trigonometric cotangent as <math display="inline">\pi \cot \pi z = \frac1z + \frac1{z-1} + \frac1{z+1} + \frac1{z-2} + \frac1{z+2} + \cdots .</math> {{pb}} {{cite journal |last=Varadarajan |first=V. S. |title=Euler and his Work on Infinite Series |journal=Bulletin of the American Mathematical Society |series=New Series |volume=44 |number=4 |year=2007 |pages=515–539 |doi=10.1090/S0273-0979-07-01175-5 |doi-access=free }}</ref>
<math display="block">
\pi \coth \pi z = \lim_{n\to\infty}\sum_{k=-n}^n \frac{1}{z + ki}.
</math>

Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the {{mvar|ni}} power is:
<math display="block">x^{n i} = \cos(n\ln x) + i \sin(n\ln x ).</math> 

Because the exponential is periodic, its inverse the [[complex logarithm]] is a [[multi-valued function]], with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of {{math|2''πi''.}} One way of obtaining a single-valued function is to treat the codomain as a [[cylinder]], with complex values separated by any integer multiple of {{math|2''πi''}} treated as the same value; another is to take the domain to be a [[Riemann surface]] consisting of multiple copies of the complex plane stitched together along the negative real axis as a [[branch cut]], with each branch in the domain corresponding to one infinite strip in the codomain.<ref>{{Cite book |last=Gbur |first=Greg |author-link=Greg Gbur |year=2011 |title=Mathematical Methods for Optical Physics and Engineering |publisher=Cambridge University Press |url=https://www.worldcat.org/oclc/704518582 |isbn=978-0-511-91510-9 |pages=278–284|oclc=704518582 }}</ref> Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.

For example, if one chooses any branch where <math>\ln i = \tfrac12 \pi i</math> then when {{mvar|x}} is a positive real number,
<math display=block> \log_i x = -\frac{2i \ln x }{\pi}.</math>

=== Factorial ===
The [[factorial]] of the imaginary unit {{mvar|i}} is most often given in terms of the [[gamma function]] evaluated at {{math|1 + ''i''}}:<ref>{{cite journal |last1=Ivan |first1=M. |last2=Thornber |first2=N. |last3=Kouba |first3=O. |last4=Constales |first4=D. |title=Arggh! Eye factorial . . . Arg(i!) |journal=[[American Mathematical Monthly]] |volume=120|pages=662–665  |year=2013 |issue=7 |doi=10.4169/amer.math.monthly.120.07.660|s2cid=24405635 }} {{pb}} [[Neil_Sloane|Sloane, N. J. A.]] (ed.).  "Decimal expansion of the real part of i!", Sequence {{OEIS link|A212877}}; and "Decimal expansion of the negated imaginary part of i!", Sequence {{OEIS link|A212878}}. ''The [[On-Line Encyclopedia of Integer Sequences]]''. OEIS Foundation.</ref>

<math display=block>i! = \Gamma(1+i) = i\Gamma(i) \approx 0.4980 - 0.1549\,i.</math>

The magnitude and argument of this number are:<ref>[[Neil_Sloane|Sloane, N. J. A.]] (ed.).  "Decimal expansion of the absolute value of i!", Sequence {{OEIS link|A212879}}; and "Decimal expansion of the negated argument of i!", Sequence {{OEIS link|A212880}}. ''The [[On-Line Encyclopedia of Integer Sequences]]''. OEIS Foundation.</ref>

<math display=block>
|\Gamma(1+i)| = \sqrt{\frac{\pi}{ \sinh \pi}} \approx 0.5216, \quad
\arg{\Gamma(1+i)} \approx -0.3016.
</math>

==See also==
* [[Hyperbolic unit]]
* [[Right versor]] in quaternions

==Notes==
{{notelist}}

== References ==
{{reflist|25em}}

== Further reading ==
*{{cite book |first=Paul J. |last=Nahin |title=An Imaginary Tale: The story of {{mvar|i}} [the square root of minus one] |location=Chichester |publisher=Princeton University Press |year=1998 |isbn=0-691-02795-1 |via=Archive.org |url=https://archive.org/details/imaginarytales00nahi |url-access=registration}}

==External links==
* {{cite web |first=Leonhard |last=Euler |author-link=Leonhard Euler |title=Imaginary Roots of Polynomials |url=http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2245&bodyId=2439 |access-date=29 November 2012 |archive-date=16 December 2019 |archive-url=https://web.archive.org/web/20191216104926/https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-factoring-rational-functions |url-status=dead }} at {{cite web |title=Convergence |website=mathdl.maa.org |publisher=Mathematical Association of America |url=http://mathdl.maa.org/convergence/1/ |url-status=dead |archive-url=https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/ |archive-date=2007-07-13}}

[[Category:Complex numbers]]
[[Category:Algebraic numbers]]
[[Category:Quadratic irrational numbers]]
[[Category:Mathematical constants]]