Both the negative sign symbol and subtraction's minus symbol look the same, so how does anyone tell them apart?
[Math] How to distinguish the difference between a negative sign and a minus sign in Algebra
arithmetic
arithmetic
Both the negative sign symbol and subtraction's minus symbol look the same, so how does anyone tell them apart?
Best Answer
The same way we distinguish different meanings of ‘$\cdot$’, for instance (e.g., $x\cdot y$, meaning ordinary multiplication of real numbers $a$ and $b$; $\vec a\cdot\vec y$, meaning the dot product of vectors $\vec x$ and $\vec y$; $\|\cdot\|$, in which the dot is a place-holder): by context. It’s no different from dealing with the problem in English of distinguishing various quite different meanings of strike, including ‘a labor action’ and the baseball player’s strike that occurs when he swings and does not manage to strike the ball.
Everyday language is full of words with multiple meanings, sometimes even contradictory meanings (cleave ‘to cut something’ and cleave ‘to adhere’). Mathematical language may perhaps suffer a bit less from such overloading, but it’s far from immune: just look at the overloading of the words regular and normal. The same goes for mathematical notation; the use of a single symbol for unary and binary minus is just one of the most familiar examples. And in all of these settings the answer to your question is that we rely on the context to disambiguate the word or symbol. This generally works quite well, especially in mathematics; occasionally it fails.
In the specific case under discussion, the most relevant contextual clue is simply the number of operands associated with the $-$: if there are two, as in $a-b$, it’s the binary operator of subtraction; if there is just one, as is clear from the parenthesis in $a(-b)$, it’s the unary operator of taking the additive inverse.