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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 are the vertical axis.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers.It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.
The smart BASIC for iOS naturally supports complex numbers in notation a + bi. Any variable, math operation or function can accept both real and complex numbers as arguments and return real or complex numbers depending on result. For example the square root of -4 is a complex number:
These definitions carry through just fine, even if the Grassmann numbers use the real numbers as the base field; however, in such a case, many coefficients are forced to vanish if the number of generators is less than 4. Thus, by convention, the Grassmann numbers are usually defined over the complex numbers.
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. [1] Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts ...
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