Imaginary unit

The name imaginary is generally credited to René Descartes, and Isaac Newton used the term as early as 1670.

The imaginary unit i is defined solely by the property that its square is −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 i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i2 with −1).

In the complex plane, which is a special interpretation of a Cartesian plane, i is the point located one unit from the origin along the imaginary axis (which is perpendicular to the real axis).

Also, despite the signs written with them, neither +i nor −i is inherently positive or negative in the sense that real numbers are.

Using the concepts of matrices and matrix multiplication, complex numbers can be represented in linear algebra.

Then a complex number a + bi can be represented by the matrix aI + bJ, and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.

More generally, any real-valued 2 × 2 matrix with a trace of zero and a determinant of one squares to −I, so could be chosen for J.

Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra.

Polynomials whose coefficients are real numbers form a ring, denoted

an algebraic structure with addition and multiplication and sharing many properties with the ring of integers.

In the geometric algebra of the Euclidean plane, the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector part.

The quotient of any two perpendicular vectors of the same magnitude, J = u/v, which when multiplied rotates the divisor a quarter turn into the dividend, Jv = u, is a unit bivector which squares to −1, and can thus be taken as a representative of the imaginary unit.

In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.

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

However, great care needs to be taken when manipulating formulas involving radicals.

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:[7]

[8][9][10] When x or y is real but negative, these problems can be avoided by writing and manipulating expressions like

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.

When multiplied by −i, any arbitrary complex number is rotated by a quarter turn clockwise.

Written as a special case of Euler's formula for an integer n,

These are the vertices of a regular polygon inscribed within the complex unit circle.

The complex exponential is thus a periodic function in the imaginary direction, with period 2πi and image 1 at points 2kπi for all integers k, a real multiple of the lattice of imaginary integers.

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

[13] The quotient coth z = cosh z / sinh z, with appropriate scaling, can be represented as an infinite partial fraction decomposition as the sum of reciprocal functions translated by imaginary integers:[14]

Other functions based on the complex exponential are well-defined with imaginary inputs.

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 2πi.

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 2πi 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.

The factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at 1 + i:[16]