# Why is $-a+b$ always equal to $b-a$?

Can someone elaborate on what rules underlie the rule

$$-a+b=b-a\;?$$

Is it associative properties of multiplication and addition? For example,

$$-a+b=b-a$$

$$(-1)(a)+b=b-a$$

$$b+(-1)(a)=b-a$$

$$b-a=b-a$$

Are there any exceptions to $$-a+b=b-a$$?

• I would say commutativity of addition and definition of subtraction as addition of the additive inverse Feb 12 '19 at 3:41

We don't have to invoke any properties of multiplication to explain why $$-a+b=b-a.$$ We don't have to invoke associativity (which refers to different ways of grouping three operands) either. We merely have to know that addition is commutative (i.e., $$x+y=y+x$$) and every element $$z$$ has an additive inverse ($$-z$$) such that $$z+ (-z) = 0.\;$$ We define subtraction by $$x-z=x + (-z)$$. In any system where those assumptions hold, it is always true that $$(-a) + b = b + (-a) = b -a$$.
$$x+y = y+x$$.
Set $$x = -a$$ and $$y=b$$.