1

It seems to me that, by convention, the equality $+x = x$ holds for any real (or complex) number $x$.

I have not, however, found such a convention explicitly presented in any text.

Well... am I wrong and there is no such convention?

Gary
  • 31,845
Paulo Argolo
  • 4,210
  • 7
    Well if $+x$ isn't $x$ by definition, then what else would it be? What prompted you to ask this question? – user829347 Jan 26 '22 at 23:55
  • Well... I still haven't found in the literature that, by definition, +x = x. However, I've seen things like this: +(+2) = 2 , +(-5) = -5, etc. Since x and y are real numbers, given the equation |x| = |y|, it is customary to write: x = ± y, that is, x = +y = y or x = -y. – Paulo Argolo Jan 27 '22 at 00:30
  • 2
    @PauloArgolo You can think of $0+2=+2$ as a notation symmetric to $0-2=-2$. – dxiv Jan 27 '22 at 00:53
  • 1
    In my opinion the reason $+x$ never needs to be defined (other than that it's obvious) is that it's not actually used but in cases where one wishes to emphasize that we are speaking of $+$ and not $-$. If you were to write a computer verifiable proof you would omit the $+$, it is used not for a mathematical purpose but for a human purpose, for emphasis. (As for the $\pm$ symbol, $\pm x$ is defined to mean both options: $x$ and $-x$. Which according to the convention, you can also write as $+x$ and $-x$ if you wish.) – Snaw Jan 27 '22 at 00:56
  • 2
    I don’t think that I would ever write “$+x$” by itself. It’s probably hard to find this explicitly defined because mathematicians never actually write this. – littleO Jan 27 '22 at 00:56
  • @PauloArgolo But why would there be any need to formally define '$+x$'? As Snaw says, it is just used for emphasis – user829347 Jan 27 '22 at 01:02
  • One instance where this notation commonly occurs is when working with the extended reals and one may write something like $\sup_{x\in\mathbb{R}}f(x)=+\infty$, but again, one just as often writes $\sup_{x}f(x)=\infty$. – JWP_HTX Jan 27 '22 at 01:09
  • You can say that $+x = (+1)(x) \Leftrightarrow +x = x$. – soupless Jan 27 '22 at 01:21
  • @VeryForgetfulFunctor "What else would it be?" Some beginners may think that $+x$ means what mathematicians write as $|x|$. – GEdgar Jan 27 '22 at 01:34
  • @dxiv Yes . I've already found in the literature -x as shorthand for 0 - x , as well as +x for short for 0 + x. – Paulo Argolo Jan 27 '22 at 01:42

1 Answers1

2

Interesting question. You can think of this as a statement about the identity. That is define the identity function $$ +x = (+1)\, x = \iota(x) := x $$ parallel to the negation function $$ -x = (-1)\, x = \nu(x) := -x. $$

It can also be used for consistency and emphasis in equations such as $$ -y + x = +x - y. $$ There may be more reasons for it being used but you are correct that

I have not, however, found such a convention explicitly presented in any text.

Perhaps the reason is that it is so obvious that it seems not to be needed to state explicitly. However, in the context of computer-aid proof and computation, in order that it will be correctly understood and processed, there needs to be explicit rules to recognize such expressions but they will be used internally and not likely to be explicitly documented as such.

Somos
  • 35,251
  • 3
  • 30
  • 76