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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.

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    $\begingroup$ $-x=0-x$, so the unary is a special case of the binary. $\endgroup$ – Matt Samuel Feb 16 at 21:52
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    $\begingroup$ @MattSamuel, they are related but they are different functions as they have different domains. $\endgroup$ – Zuriel Feb 16 at 21:55
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    $\begingroup$ Your suggestion would never fly with mathematicians. Much worse abuses of notation are in common use for ease of readability. In most cases it would make it much harder to read. It's not worth it. $\endgroup$ – Matt Samuel Feb 16 at 22:14
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    $\begingroup$ In the notation category, your question is a refreshing change of pace from the "How do I express this simple English phrase in obscure mathematical symbols?" type of question. $\endgroup$ – Robert Soupe Feb 17 at 6:02
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    $\begingroup$ There also is a tiny notational difference in professionally typeset mathematics: Look at $0 - 1$ and $-1$. In the case of a binary operator, there is a small space on both sides of the operator whereas the unary operator is written directly next to its operand. (TeX does this automatically.) I would say that the size of this distinction is about proportional to its importance. $\endgroup$ – Eike Schulte Feb 17 at 15:53
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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 positively atrocious to behold and cumbersome to write and typeset.

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    $\begingroup$ And there I sat, thinking only Shakespearean timers used "betwixt" :) $\endgroup$ – Arnaud Mortier Feb 17 at 9:05
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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., $-8 = 0 - (0 + 8)$.

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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.

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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).

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    $\begingroup$ Well, my complex analysis professor, who was from another country (though I never found out which), objected to reading $-x$ as "negative $x$" because even if $x$ is a real number it could already be negative, so $-x$ could be positive. He instructed us to say "minus $x$." $\endgroup$ – Matt Samuel Feb 16 at 22:31
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    $\begingroup$ Actually, the equation $0-6=-6$ is pronounced "zero minus six equals minus six". I believe the uncouth reading "negative six" was invented recently (last 70 years) by educrats, not mathematicians. $\endgroup$ – bof Feb 16 at 22:57
  • $\begingroup$ @MattSamuel That's a different question, I would be tempted to interpret the $-$ in $-x$ as a third different version. But that's more philosophically than mathematically, I wouldn't teach it like this. In practice, I tend to agree with your teacher. But when you need to write $-x$, usually you're already very familiar with negative numbers and you have other kind of issues. $\endgroup$ – Arnaud Mortier Feb 16 at 23:02
  • $\begingroup$ @bof I learned to read it "negative six" when I started teaching in Ireland, not being a native speaker myself. Almost all the students were reading it this way and I actually found it quite nice. $\endgroup$ – Arnaud Mortier Feb 16 at 23:08
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    $\begingroup$ As a real oldster, I must say that I stand with @bof. I grew up in New York, and was taught to read “$-6$” as “minus six”. My memory may be off, but I believe I first heard “negative six” in the Seventies or so, from college students in my classes. My impression is that professional mathematicians regularly say “minus six”, though I may be mistaken here. $\endgroup$ – Lubin Feb 17 at 6:32

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