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based on an earlier version of the Wolfram Language.
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Complex Numbers
Mathematica has fundamental support for both explicit complex numbers and symbolic complex variables. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full generality.
x+Iy the complex number x+ImaginaryI y
I (ImaginaryI) — (entered as Esc ii Esc "imaginary i", or Esc jj Esc)
Complex convert a pair of reals to a complex number
    
Re real part
Im imaginary part
Abs absolute value
Arg argument (phase angle in radians)
Sign normalized direction (z/LeftBracketingBarzRightBracketingBar)
Conjugate complex conjugate z* (also entered with superscript Esc co Esc)
ConjugateTranspose Hermitian conjugate of a matrix (also entered with Esc ct Esc)
    
ComplexExpand expand symbolic expressions into real and imaginary parts
PowerExpand expand symbolic expressions ignoring branch cuts
ExpToTrig, TrigToExp convert between complex exponentials and trig functions
    
GaussianIntegers option for polynomial and number theory functions
Reduce reduce equations and inequalities over complex numbers
    
RandomComplex random complex number
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