Should we distinguish the minus sign from the negative sign?

Question

In the set $\mathbb{C}$ of complex numbers, the minus sign "-" may be used for following:

1. As a unary operator $-_u$, given a complex number $a$, $-_ua$ is the unique number (called the negative of a) $c$ such that $a+c=c+a=0$, where $0$ is the additive identity.
2. As a binary opeartor $-_b$, given two complex numbers $a$ and $d$, $a-_bd$ is the sum of $a$ and $-_ub$.

Though $-_u$ and $-_b$ are closely related, they are completely different objects in mathematics: $_u$ is considered as a map from $\mathbb{C}$ to $\mathbb{C}$ while $-_b$ is a map from $\mathbb{C}\times\mathbb{C}$ to $\mathbb{C}$. As maps, they have the same codomain but different domains.

In some calculators such as TI nspire, two different keys are used for the two different meanings of $-$. However, in our everyday writing, we seldom differentiate the unary opeartor $-_u$ and $-_b$. Shouldn't we use diffrent symbols for them; after all, though closely related, they are different?

If we do not use different symbols for them, the following simple calculation appears confusing:

\begin{equation*}\begin{array}{c} \phantom{\times9}-23\\ \underline{-\phantom{9}-15} \\ \phantom{999}-8 \end{array}\end{equation*}

Also, I find the idea that in the same equation $-1-1=-2$, the first $-$ has different meaning from the second $-$ unsatisfactory.

Answer 1

First, note that there is already a distinction that is made, in the sense that you don't pronounce them the same: $$0-6=-6\qquad \text{"zero } \textbf{minus}\text{ six}=\textbf{negative}\text{ six}"$$ while other languages wouldn't - French, for instance, has "zéro moins six = moins six".

Now perhaps there is one case where it could be relevant to also point out the difference as you're writing, and that's when you're teaching at low level. Pupils get confused about the rules involving the $-$ sign partially because of this intentional mix-up by mathematicians and teachers (another, though related, source of confusion being the fact that you systematically write down the $-$ attached to negative numbers, while positive numbers don't have to have a $+$ at all times).

Answer 2

In my opinion, the notation should be different—but I accept that it’s not.

For another (rather egregious) example, consider the equation $$(a+bi)+(c+di) = (a+c)+(b+d)i.$$

Notice that the symbol ‘+’ serves three distinct purposes: 1) to separate the real and imaginary parts of each complex number; 2) to indicate complex addition; and 3) to indicate real number addition (right hand side of the equation).

I alert my students to this and when, inevitably, they ask why we use such deficient notation, I tell them that, with respect to notation, mathematicians try to strike a balance betwixt clarity and readability; for instance, the following substitute for the previous equation (which I’m making up for convenience) $$(a \oplus bi)+_{\mathbb{C}}(c\oplus di) = (a+_ \mathbb{R} c ) \oplus (b+_ \mathbb{R} d)i$$ contains no ambiguity but is visually unpleasant and cumbersome to write and typeset.

Answer 3

Only when there is the sort of ambiguity that could plausibly cause someone to completely misunderstand what you mean.

For example, what do I mean by $\pi(-\pi - 1)$? Do I mean $-\pi - \pi^2 \approx -13.011197$? Or do I mean 0? Or maybe I even mean 2.

The ambiguity here arises in part out of choosing to interpret the first "$\pi$" as the prime counting function or choosing to interpret it as that famous transcendental number.

Then there might be the ambiguity of whether, given positive $x$, is $\pi(-x) = 0$ or is it $\pi(-x) = \pi(x)$? In other words, is $\pi(x)$ how many positive primes there are between 0 and $x$ or is it how many primes, positive or negative, there are between 0 and $x$? The latter interpretation justifies $\pi(-4.14159) = 2$ rather than 0.

Though I suppose it would be possible to interpret both instances of $\pi$ as the prime counting function, in which case $\pi(-\pi(-1)) = 0$ regardless.

Try these in Wolfram Mathematica or Wolfram Alpha:

• Pi(-Pi - 1)
• PrimePi[-Pi - 1]
• PrimePi[-PrimePi[-1]]

This is not to say that confusion around the meaning of "$-$" is completely impossible. It is unlikely, though.

For example, we could assert both instances of "$-$" are unary negation. To make Mathematica see it that way, we'd need to do either Pi((-Pi)(-1)) or PrimePi[(-Pi)(-1)] (the former is $\pi^2$, the latter is 2). How likely is that, though? Without your question, not very.

So in general, there is no ambiguity whatsoever between negation and subtraction. However, if you like, you can choose to view negation as subtraction with an implied minuend of 0, e.g., $-8 = 0 - (0 + 8)$.

Answer 4

We do distinguish most of the time, at such a subconscious level that it seems either automatic or unthinking. Heck, we're even capable of resolving meanings that a computer would find contradictory.

Take for instance $$\frac{3}{2} = 1 + \frac{1}{2}.$$ In a cake recipe, you might find something like

1-1/2 cups light brown sugar

Put that in the context of JavaScript and it could be misunderstood as $$1 - \frac{1}{2} = \frac{1}{2}.$$ But if the author had meant one half rather than three halves, why didn't they just write one half? We understand in the recipe context that the author meant three halves, not one half.

If that's not manly enough an example, try a search for a 1-1/2 socket wrench.