# Adding degrees to complex numbers

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?

• I'm familiar with the basics of Mathematica but have never used it to manipulate complex numbers before. – George Jun 5 '19 at 18:12

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

• Or using exact numbers 1/2*Exp[I*45*Degree] // ComplexExpand – Bob Hanlon Jun 5 '19 at 18:48