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I have some conceptual issues with aforementioned definitions.

How is exactly multiplicative inverse defined? Say, for a rational, nonzero number a/b, its reciprocal is b/a. Is there a certain operation that transforms a/b to b/a?

Also, the notation for multiplicative inverse of any real number (except zero) x is 1/x. Is 1/x a unique symbol or one that indicates operation of division of 1 by x?

For example, if x=2/3, should i see its inverse as 1/x=3/2, or as an operation of division, that is 1/x=1/(2/3)? I know that in the end the answer is the same, but what i'd like to know is if division is included in the "process" of obtaining that inverse or is it by definition that we just "flip" the numbers.