Is There a Set Precedence Between Negative Numbers and Subtraction?

[C0nfused]

Hi everybody,
I would like to know your opinion about this:

The (-) minus sign is used to represent both negative numbers and subtraction. Of course, subtraction is a special case of addition, but we definitely use this operation instead of always writing a+(-b) for example. So my question is, when we write -a-b we mean this: (-a)-b? I mean, the first minus is to indicate a negative number while the second subtraction? Also, have we set any precedence between these 2 different uses of the minus sign? I know that we say that we do subtractions from left to right but a small "problem" arises when the first term of an expression is negative. When we have for example -1+2a+3-4(z^2) ... all minus signs can be thought of as subtraction except from the first one. Is it clear that it is "part" of the first number,in other words a unary operator, and not a binary operator? I guess we could solve that "problem" by saying that we mean this: 0-1+... But I am not sure which is the most sensible explanation.

And one more thing: is it because of the definition of subtraction that we say it has the same precedence as addition? And when we write a-b+2c+5 for example, is it because of this same precedence that this expression is equal to a+(-b)+2c+5 and not to a+[-(b+2c+5)]?

That's all. Sorry if all these sound stupid to u.

Thanks

 

[Moo Of Doom]

I was taught that there were actually 3 meanings of the symbol "-". The symbol can be part of a number like -5, it can mean subtraction like 4-2, or it can be negation as in -(x+2). These three cases, however are sufficiently related, that often the line blurs between them (like does - in $-\sqrt{2}$ express the negation of the result of the root, or is it simply part of the number "negative root 2"?). The ambiguity is irrelevant to the result, though, which is why we use the same symbol for all of these things.

 

[JasonRox]

I just look at the negative sign as the additive inverse of the value it is in "front" of.

a - b = a + (-b) ... the additive inverse of b.

- a - b = (-a) + (-b) ... the additive inverse of a and b.

When you having something like...

-(a + b), we say it's equal to ... -(a + b) = - a - b = (-a) + (-b) ... which is the additive inverse of (a + b).

It doesn't mean two different things in my books.

 

[C0nfused]

Thanks for your answers.
I agree (of course!) with everything you mention but I insist on my question: (-) minus sign is defined both as a unary operator and a binary one. So have we defined that these 2 uses of it have the same precedence or does this result from the definition of subtraction and of the number "-a"?

Thanks again

 

[Moo Of Doom]

Yes and no. The definition of a - b is a + (-b). But the "precedence" (or lack thereof, I guess) is purely convention. Whether we write a - b + c to mean (a - b) + c or a - (b+c) cannot be derived from any axioms, same as how a - b might as well mean -a + b (but it doesn't). The fact that multiplication precedes addition is also convention. a*b+c*d does not mean (a*b)+(c*d) because that can be deduced from the axioms, but simply because of the way we write math. The convention could well have been +(*(a,b),*(c,d)) to mean ab + cd, which is a lot clearer from the perspective of order of operations as it forces parentheses on you.

So, in short, the answer is that we defined that these 2 uses have the same precedence.