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this is used in almost every area of modern life - electrical and sound engineering, sound and image compression technology, radio transmission technologies, etc
in electrical engineering to represent the phase change component of
impedance which is a complex number containing resistance and phase components - see
frequency response analyzers
in electromagnetism where electric and magnetic fields, as well as electromagnetic waves, are modeled using complex notation for phase relationships
in quantum mechanics where the wave functions and probabilities are described using complex numbers
in fluid dynamics where they are used to describe potential flows and streamline functions in two-dimensional systems
in geometry and fractals where they are used to construct fractals (e.g., Mandelbrot set) and for rotations or transformations in the plane
for solving polynomials as every non-constant polynomial equation has a solution in the complex plane, making mathematics complete in this context
in cryptography were some encryption algorithms use complex number concepts
in graphics and simulation where image rotation, 2D/3D rendering, and visualizing data or complex functions all utilize complex numbers
in artificial intelligence where neural networks and data structures may be expressed via complex numbers for certain algorithms
in financial modeling where they are used for analyzing trends, stock prices, and economic variables
population dynamics and neuronal networks: Models population changes and neural activity using systems involving complex numbers
etc