Take the 2-minute tour ×  When I followed an introductory!
course group theory and throughout all my Math courses as a
physicist, subtraction was always defined in terms of the
inverse element and addition.  Is this the only way? I.e.: can
subtraction be defined without addition?  If this is too broad a
question, perhaps focus on groups and other physically relevant
concepts.  share|improve this question Here's teh start of a
bunch of C++ programmers discussing the question:
chat.stackoverflow.com/transcript/message/4935773#4935773, where
I held that I believe that the axioms of math are simpler if
subtraction is defined in terms of addition, but I'm not a math
guy, so we brought it here. –  Mooing Duck Aug 14 '12 at 18:00
Addition is more fundamental in the sense that the natural
numbers (the most fundamental set of numbers) is closed under
addition but not subtraction. –  Austin Mohr Aug 14 '12 at 18:02
And appending the negative numbers and zero to the natural
numbers obviates subtraction: $a-b = a+(-b)$ –