Argand diagram, representing the complex plane. Despite the historical nomenclature “imaginary”, complex numbers are regarded in the mathematical sciences as just as “real” as the real numbers, and are fundamental in many aspects complex variables theory and applications kasana pdf our description of the natural world. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.

Most importantly the complex numbers give rise to 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.

Complex numbers allow solutions to certain equations that have no solutions in real numbers. Complex numbers provide a solution to this problem. According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. An illustration of the complex plane.

From this definition, complex numbers can be added or multiplied, using the addition and multiplication for polynomials. William Rowan Hamilton introduced this approach to define the complex number system.

These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form. A position vector may also be defined in terms of its magnitude and direction relative to the origin.