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so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
so that real numbers are on the x-axis and imaginary numbers are on the y-axis.
Traditionally, mathematical objects are derived from [[physics]] or [[geometry]]. The complex numbers on the other hand were constructed out of the blue as an ad-hoc solution to certain problems that appeared to be unsolvable with real numbers. Initially unanimously rejected by the leading mathematicians, they were eventually accepted by a growing number of mathematicians due to a series of spectacular results. Not all of those results can be proved with [[elementary techniques]] however, and are thus considered questionable by more rigorous mathematicians.
Much of complex analysis is devoted to studying [[holomorphic functions]] that are infinitely differentiable. These functions take complex values in the complex plane and are differentiable as complex functions.
Much of complex analysis is devoted to studying [[holomorphic functions]] that are infinitely differentiable. These functions take complex values in the complex plane and are differentiable as complex functions.