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In complex numbers, the real part is the component that can be found on the horizontal axis of the complex plane, while the imaginary part is the component on the vertical axis, represented by a multiple of the imaginary unit 'i'. Together, these parts allow complex numbers to be expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.
The Cauchy-Riemann equations are a set of two partial differential equations that provide necessary and sufficient conditions for a complex function to be holomorphic, meaning it is complex differentiable at every point in its domain. These equations link the real and imaginary parts of a complex function, ensuring the function is conformal and preserves angles locally.
Arithmetic operations with complex numbers involve manipulating numbers in the form of a+bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. These operations extend familiar arithmetic to the complex plane, allowing addition, subtraction, multiplication, and division while considering both real and imaginary components.
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