transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
A complex number is a number of the form , where and are real numbers and is an imaginary unit. The set of complex numbers (in fact a field and topological vector space) is denoted or .
This can be thought of as:
We think of as a subset (in fact -vector subspace) of by identifying with . is equipped with a -linear involution , called complex conjugation, that maps to . Concretely, .
Complex conjugation is the nontrivial field automorphism of which leaves invariant. In other words, the Galois group is cyclic of order two and generated by complex conjugation. also has an absolute value:
notice that the absolute value of a complex number is a nonnegative real number, with
Most concepts in analysis can be extended from to , as long as they do not rely on the order in . Sometimes even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in is important, often it is enough to order the absolute values of complex numbers. See ground field for some of the concepts whose precise definition may vary with the choice of or (or even other possibilities).
The automorphism group of the complex numbers, as an associative algebra over the real numbers, is Z/2, acting by complex conjugation.
See also at normed division algebra – automorphism.
Over other subfields, the automorphism group may be considerably larger. Over the rational numbers, for instance, has transcendence degree equal to the cardinality of the continuum, i.e., there is an algebraic extension with . Any bijection induces a field automorphism which may be extended to an automorphism of over . Therefore the number of automorphisms of is at least (and in fact at most this as well, since the number of functions is also ).
See also at automorphism of the complex numbers.
The complex numbers form a plane, the complex plane. Indeed, a map given by sending to the standard real-valued coordinates on this plane is a bijection. Much of complex analysis can be understood through differential topology by identifying with , using either and or and . (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)
It is often convenient to use the Alexandroff compactification of , the Riemann sphere . One may think of as ; functions valued in but containing ‘poles’ may be taken to be valued in , with whenever is a pole of .
exceptional spinors and real normed division algebras
On the history of the notion of complex numbers:
Jean-Pierre Tignol, p. 19 of: Galois’ Theory of Algebraic Equations, World Scientific (2001) [doi:10.1142/4628]
Orlando Merino, A Short History of Complex Numbers (2006) [pdf, pdf]
Leo Corry, A Brief History of Numbers, Oxford University Press (2015) [ISBN:9780198702597]
See also:
Wikipedia, Complex numbers
Tom Leinster, Objects of categories as complex numbers, arXiv:math/0212377
Last revised on April 22, 2023 at 16:59:12. See the history of this page for a list of all contributions to it.