Is there a sage command to compute complex numbers?

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I get:

sage: z = 1 + I
sage: sqrt(1/z)
sqrt(-1/2*I + 1/2)

If you want the real and imaginary parts, you may do:

sage: s = sqrt(1/z)
sage: s.real().simplify()
1/4*2^(3/4)*sqrt(sqrt(2) + 2)
sage: s.imag().simplify()
-1/4*2^(3/4)*sqrt(-sqrt(2) + 2)

A couple things to complete the (exact) answer of Sebastien

• sqrt(1/z) will give you ONE solution of the equation t^2==1/(1+I). There are two :

  sage: solve(t^2==1/(1+I),t)

  [t == -sqrt(-1/2I + 1/2), t == sqrt(-1/2I + 1/2)]

• You may have a better grasp of the meaning of the answer(s) by asking maxima.polarform(sqrt(1/z)) and maxima.polarform(1/sqrt(z)) respectively :

  sage: maxima.polarform(sqrt(1/z))

  %e^-((%i*%pi)/8)/2^(1/4)

  sage: maxima.polarform(1/sqrt(z))

  %e^-((%i*%pi)/8)/2^(1/4)

HTH,