TITLE: Is it possible to prove that $a-b = a+(-b)$, or is it axiomatic?
QUESTION [1 upvotes]: Recently I proved a person that the fact that the multiplication of two negatives numbers gives a positive one, at least in elementary algebra, is not a convention but it can be demonstrated using other properties, like the distributive one, proving firstly that $(-a)b = -ab.$ But now he is asking me for $a-b = a+(-b)$ and I think that is the definition of substraction, or not?

REPLY [4 votes]: Formally, the definition of "$a-b$" is
$$b+(a-b)=a$$
that is, $a-b$ is the number which added to $b$ gives $a$. It is easy to argue that this definition defines uniquely a number.
To prove that $a-b=a+(-b)$ you have to show that 
$$b+ [a+(-b)]=a$$
thus $a+(-b)$ satisfies this definition.
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