Imaginary unit

Some imaginary units correspond to points ( x, y ) on the hyperbola xy = −1.

Using the concepts of matrices and matrix multiplication, imaginary units can be represented in linear algebra. The value of 1 is represented by an identity matrix I and the value of i is represented by any matrix J satisfying J² = −I. A typical choice is

$${\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\qquad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}\,.}$$

More generally, a real-valued 2 × 2 matrix J satisfies J² = −I if and only if J has a matrix trace of zero and a matrix determinant of one, so J can be chosen to be

$${\displaystyle J={\begin{pmatrix}z&x\\y&-z\end{pmatrix}}\,,}$$

whenever −z² − xy = 1. The product xy is negative because xy = −(1 + z²); thus, the points (x, y) lie on hyperbolas determined by z in quadrant II or IV.

Matrices larger than 2 × 2 can be used. For example, I could be chosen to be the 4 × 4 identity matrix with J chosen to be any of the three 4 × 4 Dirac matrices for spatial dimensions, γ¹, γ², γ³.

Regardless of the choice of representation, the usual rules of complex number mathematics work with these matrices because I × I = I, I × J = J, J × I = J, and J × J = −I. For example,

$${\displaystyle {\begin{aligned}J^{-1}&=-J\,,\\\left(aI+bJ\right)+\left(cI+dJ\right)&=(a+c)I+(b+d)J\,,\\\left(aI+bJ\right)\times \left(cI+dJ\right)&=(ac-bd)I+(ad+bc)J\,.\end{aligned}}}$$

The two square roots of i in the complex plane

The three cube roots of i in the complex plane

Just like all nonzero complex numbers, i has two square roots: they are[lower-alpha 1]

${\displaystyle \pm \left({\frac {\sqrt {2}}{2}}+{\frac {\sqrt {2}}{2}}i\right)=\pm {\frac {\sqrt {2}}{2}}(1+i).}$

Indeed, squaring both expressions yields:

${\displaystyle {\begin{aligned}\left(\pm {\frac {\sqrt {2}}{2}}(1+i)\right)^{2}\ &=\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\ \\&={\frac {1}{2}}(1+2i+i^{2})\\&={\frac {1}{2}}(1+2i-1)\ \\&=i.\end{aligned}}}$

Using the radical sign for the principal square root, we get:

${\displaystyle {\sqrt {i}}={\frac {\sqrt {2}}{2}}(1+i).}$

The three cube roots of i are:[9]

${\displaystyle -i,}$

${\displaystyle {\frac {\sqrt {3}}{2}}+{\frac {i}{2}},}$ and

${\displaystyle -{\frac {\sqrt {3}}{2}}+{\frac {i}{2}}.}$

Similar to all the roots of 1, all the roots of i are the vertices of regular polygons, which are inscribed within the unit circle in the complex plane.

Multiplying a complex number by i gives:

${\displaystyle i(a+bi)=ai+bi^{2}=-b+ai.}$

(This is equivalent to a 90° counter-clockwise rotation of a vector about the origin in the complex plane.)

Dividing by i is equivalent to multiplying by the reciprocal of i:

${\displaystyle {\frac {1}{i}}={\frac {1}{i}}\cdot {\frac {i}{i}}={\frac {i}{i^{2}}}={\frac {i}{-1}}=-i~.}$

Using this identity to generalize division by i to all complex numbers gives:

${\displaystyle {\frac {a+bi}{i}}=-i(a+bi)=-ai-bi^{2}=b-ai.}$

(This is equivalent to a 90° clockwise rotation of a vector about the origin in the complex plane.)

The powers of i repeat in a cycle expressible with the following pattern, where n is any integer:

${\displaystyle i^{4n}=1}$

${\displaystyle i^{4n+1}=i}$

${\displaystyle i^{4n+2}=-1}$

${\displaystyle i^{4n+3}=-i,}$

This leads to the conclusion that

${\displaystyle i^{n}=i^{(n{\bmod {4}})}}$

where mod represents the modulo operation. Equivalently:

${\displaystyle i^{n}=\cos(n\pi /2)+i\sin(n\pi /2)}$

Although we do not give the details here, if one chooses branch cuts and principal values to support it then this last equation would apply to all complex values of n.[10]

Making use of Euler's formula, ${\displaystyle i^{i}}$ has infinitely many values

${\displaystyle i^{i}=\left(e^{i(\pi /2+2k\pi )}\right)^{i}=e^{i^{2}(\pi /2+2k\pi )}=e^{-(\pi /2+2k\pi )}\,,}$

for any integer k. A common principal value corresponds to ${\displaystyle k=0}$ and gives ${\displaystyle i^{i}=e^{-\pi /2}}$, which is 0.207879576....[11][12]

The factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at ${\displaystyle 1+i}$:[13][14][15]

${\displaystyle i!=\Gamma (1+i)=i\Gamma (i)\approx 0.498015668-0.154949828i.}$[16]

The magnitude of this number is

${\displaystyle |\Gamma (1+i)|={\sqrt {\frac {\pi }{\sinh \pi }}}=0.521564046\ldots ,}$[17]

while its argument is

${\displaystyle \arg {\Gamma (1+i)}=\lim _{n\to \infty }{\biggl (}\ln {n}-\sum _{k=1}^{n}\operatorname {arccot} {k}{\biggr )}\approx -0.301640320.}$[18]

Many mathematical operations that can be carried out with real numbers can also be carried out with i, such as exponentiation, roots, logarithms, and trigonometric functions. The following functions are well-defined, single-valued functions when x is a positive real number.

A number raised to the ni power is:

${\displaystyle x^{ni}=\cos(n\ln x)+i\sin(n\ln x).}$

The ni^th root of a number is:

${\displaystyle {\sqrt[{ni}]{x}}=\cos \left({\frac {\ln x}{n}}\right)-i\sin \left({\frac {\ln x}{n}}\right)~.}$

The cosine of ni is:

${\displaystyle \cos ni=\cosh n={\frac {1}{2}}\left(e^{n}+{\frac {1}{e^{n}}}\right)={\frac {e^{2n}+1}{2e^{n}}}\,,}$

which is a real number when n is a real number.

The sine of ni is:

${\displaystyle \sin ni=i\sinh n={\frac {1}{2}}\left(e^{n}-{\frac {1}{e^{n}}}\right)i={\frac {e^{2n}-1}{2e^{n}}}i\,,}$

which is a purely imaginary number when n is a real number.

In contrast, many functions involving i, including those that depend upon log i or the logarithm of another complex number, are complex multi-valued functions, with different values on different branches of the Riemann surface the function is defined on.[19] For example, if one chooses any branch where log i = πi/2 then one can write

${\displaystyle \log _{i}x=-{\frac {2i\ln x}{\pi }}\,,}$

when x is a positive real number. When x is not a positive real number in the above formulas then one must precisely specify the branch to get a single-valued function; see complex logarithm.