Writing expressions: why not write $+-3$?

Question

When learning relative numbers, we write such things as: $$ 4+(-5)+3\times(-2)-(+6)$$

Why not write instead $$4+-5+3\times-2-+6$$

I think there's no possible ambiguity between unary minus and binary, nor any problem with precedence rules.

Answer 1

You're right, formally there is no ambiguity. However, there are a few non-formal ambiguities to resolve:

• It looks like a typo, so writing brackets lets the (proof)reader know that this was 100% intentional
• Depending on the formatting available, $2+-6$ might mean $2\pm 6$ (this was probably more important a few decades ago, but conventions stick around)
• In a bad copy / print, the multiplication sign in $2\cdot -6$ might look like a random speck of ink

Also, it looks awkward, although that might just as well be from lack of use.

Answer 2

Psychologically, it is to insist that (-x) is a number despite the fact that we use two symbols to write it (and that the first symbol is the same as that of the subtraction).

Answer 3

I don't know what you mean by "relative numbers". Negative numbers are numbers in their own right. They are the additive inverses of their positive counterpart.

Without this notion of inverse element to an operator much of group theory would be difficult to come up with. Maybe for elementary algebra you are right, it does not make much difference, but to take one step further and develop algebra more, it would actually be a mechanism designed to hold us back.