[Math] Should we distinguish the minus sign from the negative sign

binary operationselementary-functionsnotation

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.

Best Answer

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.

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