Imaginary number

A ( pure) imaginary number is a number whose square is a negative real number. Equivalent to the imaginary numbers as can be those complexes define numbers whose real part and the imaginary part of zero -zero. The term " imaginary number " probably Gerolamo Cardano of (16C) was coined. In his view, such numbers could not exist, they could therefore only be imaginary (imaginary ). The imaginary unit allows the extension of the field of real numbers to the field of complex numbers.

In electrical engineering, is used as a symbol instead of a, this term goes back to Charles P. Steinmetz. The designation is in accordance with DIN 1302, DIN 5483-3 and ISO 80000-2 allows a symbol to avoid confusion with the instantaneous value of the current in applications such as the complex AC circuit analysis.

General

With imaginary numbers can be solved, their solutions can not be a real number equations. The equation

Has to solve two real numbers, -1 and 1.

The equation

However, can not have real solution, since this root would have to be drawn from a negative real number, because the root is the inverse of squaring - and squares of real numbers are always positive. Their solutions are i and- i, two imaginary numbers.

A study of square roots of negative numbers was necessary in the solution of cubic equations in the case of casus irreducibilis.

Today we understand imaginary numbers as a special complex numbers. Every complex number can be represented as the sum of a real number and a real multiple of the imaginary unit i, a number with the property:

Algebraically, i is defined as a root of the polynomial, and the complex numbers as the thereby generated field extension. The second zero is then automatically -i. They can only distinguish, however, if one refers to one of the two i. Since they are anyway but can not distinguish it does not matter " what " is now referred to zero with i.

All complex numbers can be represented in the Gaußebene, an extension of the real line ( see figure at right ). The complex number has the real part and the imaginary part. Due to the rules of calculation of complex numbers is the square of a number whose real part is equal to 0, a non- positive real number:

Thus, the imaginary numbers form a straight line which passes through the number 0 and is perpendicular to the real line. They are real multiples of the imaginary unit i

A more detailed description can be found in the article on complex numbers.

Potencies

Powers of satisfy the following relations, with integers:

Some powers are often occurring from:

Context

Extensions represent the hypercomplex numbers, which have also consistently over the complex numbers more imaginary units per number. For example, occur in the four-dimensional quaternions three imaginary units, and in the eight-dimensional octonions seven imaginary units per number to be used.

In the Eulerian identity a concise, simple connection, the imaginary unit is manufactured with three other fundamental mathematical constants, namely the Euler's number, the wave number and the real unit 1: