Such plots are named after Jean-Robert Argand — , although they were first described by Norwegian—Danish land surveyor and mathematician Caspar Wessel — Main article: Stereographic projection Riemann sphere which maps all points on a sphere except one to all points on the complex plane It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane. We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows.
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In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Furthermore, complex numbers can also be divided by nonzero complex numbers.
Overall, the complex number system is a field. 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.
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 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 this complex number. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis.
The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space; for example, the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the scalar multiplication , the scalar product or other sesqui linear forms , available in many vector spaces; and the broadly exploited vector product exists only in an orientation -dependent form in three dimensions.