Meaning of asterisk and meaning of a filled square

While I was looking through a textbook (the name of the book and the author is indicated below), I came across the following symbols which I could not understand.

Source: Introductory Notes in Discrete Mathematics by Marcel B. Finan

I indicated the symbols with a red circle and an ellipse. The asterisk normally means multiplication. But it is used in a different sense in the solution.

Also, what is the meaning of the black-filled square? I guess that it signifies the end of solution. What do you think?

Z* seems to mean Z \ {0}. The square, I don’t know. Maybe QED?

Here you go:

As for the Square - it’s Q.E.D.

I examined this page. According to this page, asterisk could mean three things:

  1. multiplication
  2. adjoint
  3. complex conjugate

None of the above concepts seems related to the context of the solution in the image file. So, what could it be? What do you think?

It absolutely means the non-zero integers, just like Ignotus said. (which needs to be pointed out since you can’t divide by zero). There are more uses for an asterisk than those listed in the mathworld page.

In fact, it is unlikely that there will ever be a “master list of ways one can use an asterisk in a mathematical proof” that is comprehensive. Proofs and textbooks define new notations that are convenient in the context all of the time. Typically, a notation like (X)* means “something like (X), but different in some way”.

Does the text book have a page at the front or back listing the meaning of these notations (in the context of the book)? That is very common for text books.

All mathematical symbols (including pi) are multiply used. In that context it has to mean non-zero integers, but in different contexts it will mean something else. For example, if V is a vector space, V* denotes the dual space. The solid block is used standardly as an end of proof marker, although most styles would put it at the right hand margin.

To elaborate: it’s a symbol placed at the end of a proof to mean “And that completes the proof of the theorem.” Some books/publications/authors use a filled-in square for this purpose, some use an open square, some use the letters Q.E.D., some use words like “This completes the proof” or “…and thus the theorem is proved,” and some don’t use any indication at all.

Yes, and you might have to look for Z* rather than just * on that list of symbols.

FWIW, Z* would be pronounced “Z-star.”

If I was editor of a mathematical journal, required style would be to end proofs with “mic drop”.

When I see “<set>" I think it’s the closure of the set under some operator. But in the example given, I don’t see what "Z” would be used for, instead of just “Z”. (And what operator the closure is based on isn’t knowable without seeing more of the text.)

The Set of Non-Zero Integers

It’s fairly standard in my experience, but that page show Z* as ‘deprecated’.

What it indicates to be standard (ℤ[sub]≠0[/sub]) seems inherently clearer and a lot more versatile if you change the subscript. I could see the old one falling out of use just for the practicality of the new one.


Were you an editor of a mathematical journal and can’t remember? Otherwise your post should read “If I were an editor…” Which means–again, unless you are mentioning for some reason unknown to us you may have been an editor (which would be irrelevant, frankly)–you have made a mistake.



So the proof notes that q is a non-zero int in order for the following division to be valid. Got it.

(How often does one have to explicitly make it clear that a number isn’t zero before using it as a divisor in a proof? Crucial in programming, but in proofs?)

It all depends on the level. In an elementary text, you can’t do it too often. In an advanced one, you would just assume the reader will fill in the argument.

The idea that Z* is deprecated is absurd. People can deprecate away; they will be ignored. I am actually a technical editor of a free online journal and we use an open square to end proofs. But it is moved to the end of the last line.

There is no standard notation in mathematics, ever. Even pi and e are regularly used fr whatever, just so long as you don’t need those real constants. And * is used all over the place. There is even a concept I know a bit about called *-autonomous categories.

In general, proofs are sorta like programming. You want to make sure there aren’t any holes or unwarranted assumptions so that your proof doesn’t crash.

In this particular example, q being a non-zero int is part of what is being proven. That is, it’s a proof that a + b is a rational number (an element of Q), which means by definition that it can be written as p/q where p is an integer (an element of Z) and q is a non-zero integer (an element of Z*).

The “if I was” construction for hypotheticals is also usual in UK English, as far as I know. Looks like leahcim is likely American, given her or his location, but it’s not a rule that holds across varieties of English (and the distinction seems to be falling out of use slowly in American English, too.)

FWIW, Wolfram MathWorld, has a page specifically for Z* (read “zee star”). They say it excludes the negatives but keeps 0. That seems to be the exact opposite of what the OP is talking about.

Because nobody’s mentioned it: The filled square is the Halmos tombstone, or just the halmos, named after Paul Halmos, who popularized it. (He got the idea of using it from seeing it was being used to indicate the end of articles in magazines.)


I see what you did there. :slight_smile: