Complex number field
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit is a solution of the equation i2 = −1, that must be "imagined" because there is no real number solving this equation. In this expression, a is called the real part of the complex number, and b is called the imaginary part. Despite the historical nomenclature "imaginary", the existence of complex numbers is considered in the mathematical sciences to be settled. According to Roger Penrose: "complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable", adding that they are needed for the description of "the precise operations of [the natural] world at its minutest scales."
The complex number system is defined as the algebraic extension of the ordinary real numbers by an imaginary number i. This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.
The most important property of the complex numbers is the fundamental theorem of algebra: every non-constant polynomial equation with complex coefficients has a complex solution. This property is true of the complex numbers, but not the reals. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.
Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as simply as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of the complex number.
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