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The geometric identification of the complex numbers with the complex plane, which is a [[Euclidean plane]] (<math>\mathbb R^2</math>), makes their structure as a real 2-dimensional [[vector space#Complex numbers and other field extensions|vector space]] evident. 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 [[bilinear form|forms]], available in many vector spaces; and the broadly exploited ''[[vector product]]'' exists only in an [[orientation (vector space)|orientation]]-dependent form in three dimensions.
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==Definition==
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