Complex number
Complex number
The complex numbers also form a real vector space of dimension two, with ${\displaystyle \{1,i\}}$ as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of ${\displaystyle i}$ are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation) in the complex plane, and multiplying by a fixed complex number is a similarity) centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.
Complex number
The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra) over the reals, and a Euclidean vector space of dimension two.
Definition and basic operations
!Various complex numbers depicted in the complex plane.
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