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The method .N() or .n() is a shortcut for .numerical_approx()
and will try to give you a numerical approximation that lives in RR
which is a shortcut for RealField().
If you specify a precision, .N(10) or .n(10) or .numerical_approx(10)
give you a numerical approximation that lives in RealField(10). Here,
the precision 10 specifies the number of bits of precision.
You can also specify the number of (base ten) digits of precision:
by using .N(digits=10) or .n(digits=10) or .numerical_approx(digits=10).
So the answer to your question is yes, the two things you are using are doing the same thing.
Note that the fastest floating-point real numbers in Sage are RDF,
short for RealDoubleField.
So in general, I would advise to compute in RDF, use RDF's pi
sage: pi_n = RDF.pi()
and map everything you need to compute with into RDF.
See a more detailed comparison of all approximations of the real field in SageMath in this ask-sage question:
http://ask.sagemath.org/question/9950/what-are-the-different-real-numbers-in-sage/
