I am writing a scientific paper.
I am not sure about the usage of "assume."
Should I say "Let us assume x is real," or should I say "Let us assume x be real"?
Let us assume x is real
This sounds about right.
Let us assume x be real
This is grammatically incorrect.
Let us assume x to be real
This is grammatically correct, but sounds awkward, though with more context, it could be more correct than is.
Your question has made me think about this for the first time: there seems to be a distinction between verbs that merely process information about a fixed situation, and verbs that declare a position or approach to a potentially changeable situation.
For example you would say
I believe [that] X is real.
and not "I believe X be real". By contrast you should really say
I demand [that] X be real.
and not "I demand X is real".
One result of this is that the following two statements have different meaning:
I insist that X is real
means "I emphatically believe that the proposition 'X is real' describes the current situation correctly, independent of my intervention or yours", whereas
I insist that X be real
means something along the lines of "I am laying down the law, that you must ensure X is real", or more generally "X may be real or not, but if it is not real then I will refuse to proceed/cooperate with whatever we're talking about".
Your example, of assuming whether something is real, is processing information about a fixed proposition. Therefore you would not use the subjunctive. You would say "assume that X is real", not "assume that X be real".
In a mathematical paper where are you are describing your initial conditions, you can choose whether to use the metaphor of actively creating X as you define it ("let X be a real number") or the metaphor of merely describing the state of an already-defined entity ("assume X is a real number"). The former is probably more common.