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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?

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    $\begingroup$ It can serve as an axiom or setting in ring theory. $\endgroup$
    – Wuestenfux
    Oct 29, 2017 at 19:54

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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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