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I noticed the following strange scenario. When I defined a variable to be real, Mathematica does not only recognize that it is real after taking an inverse. How can I resolve this so that it will recognize 1/x is still real? 

In[29]:= $Assumptions = {x ∈ Reals}

Out[29]= {x ∈ Reals}

In[30]:= Im[x]

Out[30]= Im[x]

In[31]:= Refine[Im[x]]

Out[31]= 0

In[32]:= Refine[Re[x]]

Out[32]= x

In[33]:= Refine[Im[1/x]]

Out[33]= Im[1/x]

In[34]:= Refine[Re[1/x]]

Out[34]= Re[1/x]
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    $\begingroup$ Another way to do it is use ComplexExpand without any assumptions. For example: $Assumptions = True; ReIm[ 1 / x ] // ComplexExpand $\endgroup$
    – LouisB
    Commented Jun 6, 2019 at 22:29
  • 2
    $\begingroup$ You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find this meta Q&A helpful $\endgroup$
    – Michael E2
    Commented Jun 6, 2019 at 23:36

1 Answer 1

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If x=0, then 1/0 is ComplexInfinity. If you add the assumption that x!=0, then you get what you are wanting:

 Assuming[{b \[Element] Reals, b != 0}, Refine[Im[1/b]]]

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