Is an empty parenthesis a valid mathematical expression? [closed]

Solution 1:

We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple.

Solution 2:

Generally, no.

But you could say:

Let us denote something with ().

... and start using (), if you think this would convey your point.

There is no need for academic reference validating this, it is the matter of basic author freedom.

Let's say you want to wear a violet tie with blue dots. You don't ask if there is a law permitting you to do this.

However, I would advise you not to use it, it creates confusion.

Solution 3:

We use parentheses to indicate the order of operations.

To refer to your example: the operator $+$ takes two arguments, in the form of $a+b$. You can think of $+$ as a function that takes two variables.

In this example, the $+$ is missing the second argument: there's nothing there. And $15 + $ isn't a valid mathematical statement -- it's not equal to anything, because the $+$ operation requires two variables.

An empty pair of parentheses thus doesn't really mean anything -- it just contains nothing. This destroys any equation it is placed inside -- because $()$ is just nothing, not even $0$, and mathematical operations aren't defined on nothing.