Question about using the notation "$\pm$" to represent the answer briefly.

Question

I saw this problem in the book with answer:

Solve the equation $z^4+z^2+1=0$.

The book gave the final answer as: $$z_1=\frac{1}{2}+ \frac{\sqrt{3}}{2}i, \quad z_2=\frac{-1}{2}- \frac{\sqrt{3}}{2}i,\quad z_3=\frac{-1}{2}+ \frac{\sqrt{3}}{2}i,\quad z_4=\frac{1}{2}- \frac{\sqrt{3}}{2}i.$$

My question isn't about how to solve the equation, But I want to know Is it possible to write the roots of the equation briefly like: $$z=\pm \frac{1}{2}\pm \frac{\sqrt{3}}{2}i. $$

And does it represent the same $z_1$ , $z_2$ , $z_3$ , $z_4$ I mentioned earlier? And finally I want to know whether this is a common way to represent four answers or not.

Answer 1

From G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers (4th ed. 1960), p.193:

Theorem 229. If $\xi^3 + \eta^3 + \zeta^3 = 0,$ then one of $\xi, \eta, \zeta$ is divisible by $\lambda.$

Let us suppose the contrary. Then $$ 0 = \xi^3 + \eta^3 + \zeta^3 \equiv \pm1\pm1\pm1 \pmod{\lambda^4}, $$ and so $\pm1 \equiv 0$ or $\pm3 \equiv 0 \ [\ldots].$

Surely if Hardy thinks it's OK, then it's OK.

Answer 2

There are, I think, two questions implicit in the original post:

1. Is it acceptable to write $\pm a \pm b$?

   I think that the answer to this is a qualified "yes". As noted by Calum Gilhooley, there are notable texts and authors who have written $\pm a\pm b$ to denote a four element set, i.e. $$ \pm a\pm b = \{ a+b, a-b, -a+b, -a-b\}. $$ Because this notation has appeared in the literature, if you use it in your own writing, you will likely be safe from criticism.

   On the other hand, illustrious authors often get away with things that the rest of us can't, simply on the basis of their reputations. Thus I would be cautious about emulating the style of famous authors, as they are often famous for their mathematical discoveries, rather than their clear exposition or writing ability.

2. Should one write $\pm a\pm b$?

   This is a matter of opinion, but I would recommend against writing $\pm a \pm b$. This notation has the potential to be ambiguous. The notation $ \pm a \mp b $ unambiguously denotes the two-element set $$ \{a-b, -a+b\}, $$ hence it is reasonable to infer that the notation $\pm a\pm b$ ought to denote the analogous two-element set $$ \{a+b,-a-b\}. $$ However, as amWhy noted in chat one can make a very good argument that $\pm a\pm b$ should denote the four-element set $$ \{a+b, a-b, -a+b, -a-b\}. $$ As the notation has two very reasonable interpretations, it has the potential to be ambiguous and to cause confusion.

   Because the notation has the potential to be ambiguous, I think that one should be careful about when one uses it. In one's own personal notes or on a blackboard, where the style of communication is informal and intentionally abbreviated, it can be perfectly reasonable to reduce the number of symbols written, and just write $\pm a\pm b$. In these settings, there are other channels of information which disambiguate: the verbal lecture, surrounding notes, and (perhaps) the context in which the notation is used.

   On the other hand, I would suggest that, in formal writing, one really ought to avoid the notation $\pm a\pm b$. Off the top of my head, there are a few alternatives which might be better:

   • Write "The four-element set $\pm a\pm b$..." or "The two-element set $\pm a\pm b$...", depending on what you mean.

   • Similarly, in the specific case highlighted in the original question, one could write "The four roots are given by $\pm a\pm b$..."

   • Write something more explicit, such as $$ \bigl\{ (-1)^ma + (-1)^nb : m,n \in \mathbb{N} \bigr\}. $$

   • Just list out all four elements, e.g. "The solutions are $a+b$, $a-b$, $-a+b$, and $-a-b$."

   Remember that the goal of mathematical writing is to be clear and unambiguous. While one certainly can write $\pm a \pm b$, my feeling is that one shouldn't. Of course, at the end of the day, this is between the author, the reader, and (perhaps) the publisher, and one should do whatever one believes will satisfy the needs of all parties the best.

Answer 3

You can do that, but you have to be careful. Sometimes $z=\pm \frac{1}{2}\pm \frac{\sqrt{3}}{2}$ will mean the four roots you've mentioned. But sometimes it might mean just the two roots $z_1,z_2$ where the sign is the same, with the other two denoted as $z=\pm \frac{1}{2}\mp \frac{\sqrt{3}}{2}$.

I would probably write what you've written, but add a few clarifying words ("the four roots $z=\pm \frac{1}{2}\pm \frac{\sqrt{3}}{2}$") to make sure there's absolutely no confusion.

Answer 4

Yes it is an acceptable way to write that answer. On one hand, it can be ambiguous in certain situations, as sometimes people write e.g. $ a \pm b \pm c$ to represent just $a+b+c$ and $a-b-c$ only. Or similarly something like $a \pm b \mp c$ to represent $a+b-c$ and $a-b+c$ only.

On the other hand, it is fairly common practice to use that notation, and here it is fairly unambiguous as we are expecting 4 roots.

It would be nice to say something like: "there are four distinct roots, given by $\pm a \pm b$".