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I'd like to take the complex number $(1 + 0i)$ and (i) multiply it by $0.5$ and (ii) add 45 degrees to it (without changing its magnitude). But I'd like to get its complex represenation in non-polar coordinates (i.e., some $(x +yi)$).

How do I do compute like this in Mathematica?

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  • $\begingroup$ I'm familiar with the basics of Mathematica but have never used it to manipulate complex numbers before. $\endgroup$
    – George
    Commented Jun 5, 2019 at 18:12

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By adding 45 degrees do you mean rotating by 45 degrees?

Let's do this symbolically first. Multiply a complex number x + I y by a scalar a and then rotate by θ.

e1 = a (x + I y) E^(I θ)

a E^(I θ) (x + I y)

Then to get this into the x + I y form you do

ComplexExpand[e1]

a x Cos[θ] - a y Sin[θ] + I (a y Cos[θ] + a x Sin[θ])

In the numerical case this is just 

1 0.5 E^(I π/4)

0.353553 + 0.353553 I

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  • $\begingroup$ Or using exact numbers 1/2*Exp[I*45*Degree] // ComplexExpand $\endgroup$
    – Bob Hanlon
    Commented Jun 5, 2019 at 18:48

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