Editing Imaginary unit (section)

== 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>