MathbbC

mathbbC denotes the set of complex numbers, consisting of all numbers a + bi with a and b real and i^2 = -1. It forms a field under the usual addition and multiplication, and it is algebraically closed: every nonconstant polynomial with complex coefficients has a root in mathbbC. The characteristic is 0. As a real vector space, mathbbC has dimension 2 with basis {1, i}, making it isomorphic to R^2; the real numbers embed into mathbbC via a -> a. The set is uncountable, with cardinality equal to the continuum. For z = a + bi, the modulus is |z| = sqrt(a^2 + b^2) and the conjugate is z̄ = a - bi. The polar form z = r e^{iθ} (with r ≥ 0) is often used; multiplication corresponds to multiplying moduli and adding arguments. The standard metric d(z, w) = |z - w| turns mathbbC into a complete metric space, homeomorphic to R^2 and equipped with the Euclidean topology. Complex analysis studies functions from mathbbC to mathbbC that are holomorphic; entire functions are holomorphic on all of mathbbC. MathbbC is the natural field of scalars in many areas of mathematics and physics, providing the complex algebraic, geometric, and analytic framework widely used across disciplines.