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# Complex Numbers

## Eulerâ€™s formula

A useful expression is Eulerâ€™s formula which expresses an exponential with imaginary argument in terms of a sum of real and imaginary parts: We can see where this comes from using the series representations of the exponential, sine and cosine functions If we extend the exponential series expression for imaginary argument z = iθ where θ is real, then Exponential form: Using Eulerâ€™s formula, it is possible to compactly write a complex number in terms of an exponential function: On an Argand diagram, complex numbers with the same modulus r = | z | but different arguments θ make up points on a circle centred on the origin with radius r = | z |. For modulus r = 1, the circle is of radius equal to one. Special points of interest (for r = 1) are:  