Why is negative minus negative not negative? Why is negative times positive not directionless?

Question

What I understand is that the only difference between plus and minus is direction.

I've never understood this: That +1 + +1 is +2, why is -1 - -1 not -2?

(In the first, we are moving from right to the right to the right, so we get right. In the second, we are moving from left to the left to the left, so why don't we get left?)

What's a logically/ philosophically logical/ true explanation? What are the base axioms that everything is defined upon?

Indeed, why does a negative(left) number times a positive(right) number equal negative(left)? Shouldn't it have an undefined direction with respect to left–right?

Answer 1

The $\,\color{blue}{+}\,$ or $\,\color{blue}{-}\,$ sign attached to the number is part of the number itself. The $\,\color{red}{+}\,$ and $\,\color{red}{-}\,$ signs between the numbers are operations, even though (confusingly enough) we use the same symbols for both.

I've never understood this: That +1 + +1 is +2,

In light of the above, this is the same as $\,(\color{blue}{+1}) \color{red}{+} (\color{blue}{+1}) = (\color{blue}{+2}) = 2 \cdot (\color{blue}{+1})\,$.

why is -1 - -1 not -2?

Because replacing the number $\,(\color{blue}{+1}) \to (\color{blue}{-1})\,$ in the above gives $\,(\color{blue}{-1}) \color{red}{+} (\color{blue}{-1}) = 2 \cdot (\color{blue}{-1})=(\color{blue}{-2})\,$.

In fact, both are particular cases of the general identity $\,a + a = 2 \cdot a\,$, the first one for $\,a=\color{blue}{+1}\,$, and the second one for $\,a=\color{blue}{-1}\,$. In the same way, the identity $\,a-a=0\,$ gives $\,(\color{blue}{+1}) \color{red}{-} (\color{blue}{+1}) = 0\,$ and $\,(\color{blue}{-1}) \color{red}{-} (\color{blue}{-1}) = 0\,$ for $\,a=\color{blue}{+1}\,$ and $\,a=\color{blue}{-1}\,$, respectively.

Answer 2

The result of the subtraction $x - y$ of integers $x$ and $y$ is defined as the unique integer $z$ which satisfies $x = y + z$.

The $-$ sign in $-1$ is a syntactic convention to denote the inverse of 1 for addition. Using this notation in the definition above gives $0 - 1 = (-1)$, which a posteriori justifies the choice of notation.

When the sign of an integer is important, another convention is to write $+x$ for positive integers $x$. This other syntactic convention is justified by (part of) the definition of addition: $0 + x = x$.

It is tempting to interpret every $+$ or $-$ sign as denoting an addition or subtraction. To make this work, it is necessary to add a "sign rule" to deal with the (purely syntactic cases) when two sign symbols appear next to each other. That way, we can forget about the previous syntactic conventions and compute only at the syntactic level to reach the correct result.

Answer 3

I will try to answer philosophically (with math terms).

"Minus" means "retrieving", or "to add opposite things" (adding additive inverse). So if you are doing $A - A$, it means you want to add "inverse of $A$" to $A$ [written as $A+(-A)$], and philosophically it is zero (nothingness) since you are adding oneself (A) to its inverse.

Think it in another way, if you have 2 apples, and you eat both, it is $2-2$ and it becomes none. So if you loan a person 1 dollar ($-1$), and the person "cancel" (minus) the loan [by returning you money], then the mathematical equation is indeed $-1 - (-1) = 0$ since the person has cleared all loans by returning money to you.

Answer 4

A little known fact about negatives is that one does not need to use them at all - that is, to use special symbols and rules for them, as others have hinted. You can simply replace all the $-x$s with $(0 - x)$, but then you can just follow really obvious rules.

The universal origin of all the cases and questions about negatives is to subtract the greater from the less. Subtraction, which is common in all cases, not direction, is the underlying concept in negatives.

The surprise of algebra is that, when you do this, all the same rules for "whole numbers" continue to work. These rules, like the distributive property, are pretty self-evident for the numbers we totally understand which manifest themselves obviously in real life. - But they still form a consistent system when you apply them to subtractions like $2 - 4$, which is all negative numbers are.

The real rule that underlies the first thing you asked about is this: It is one of the associative properties of addition and subtraction. It can be derived with no reference at all to negatives or direction. I give it universally, with a "positive" example, then with negatives written as subtractions:

$$a - (b - c) = a - b + c$$ $$5 - (3 - 1) = 5 - 3 + 1$$ $$5 - (0 - 2) = 5 - 0 + 2$$ that is the explanation of the more mystifying: $$5 - -2 = 5 + 2$$ ...and here is your example: $$(0 - 1) - (0 - 1) =$$ Treat the first parenthetical as one thing and apply our rule: $$(0 - 1) - 0 + 1 =$$ Now reorder and simplify: $$0 - 1 - 0 + 1 = 0 - 0 + 1 - 1 = 0$$

All you need to explain the multiplication is the distributive property, also derived from positives: with positives: $$a \cdot (b - c) = ab - ac$$ $$3 \cdot (4 - 2) = 3\cdot4 - 3\cdot2 = 6$$ Now with your example: $$1 \cdot -1 = 1 \cdot (0 - 1) = (1 \cdot 0) - (1 \cdot 1) = 0 - 1$$ The trick of algebra is applying the rules of positive arithmetic to letters and symbols no matter what they mean.