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I'm curious whether there's an accepted notation for a function that takes in a number and returns only $\pm 1$ where the sign is the same as the sign of the input. What I'm trying to achieve feels similar to normalizing a vector where the "direction" is only either positive or negative.

$$ \text{sign}(n) = \frac{n}{\left|n\right|} $$

Examples being that $\text{sign}(-4) = -1$ and $\text{sign}(\pi^e) = 1$ .

I ask the question because of a problem where I'm taking a square root of number that has been squared ($\sqrt{a^2}$ where $a$ is not directly known) with the sign of $a$ being indicated by a different but related number $b$. I'm wanting, if one exists, a recognizable way of concisely writing what I've defined as $\sqrt{a^2}=\text{sign}(b) \times a$ without having to define my own notation where some may already exist.

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  • $\begingroup$ What would be the sign of $i$ with $i^2=-1$? $\endgroup$
    – Dietrich Burde
    Commented Aug 11, 2020 at 20:36
  • $\begingroup$ It hardly seems worth posting this as an answer: Sign function - Wikipedia. $\endgroup$
    – Calum Gilhooley
    Commented Aug 11, 2020 at 20:37
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    $\begingroup$ I have seen $\operatorname{sign}$ used as you describe. I have also seen $\operatorname{sgn}$ and $\operatorname{signum}$ used in this way. $\endgroup$
    – Xander Henderson ♦
    Commented Aug 11, 2020 at 20:37
  • $\begingroup$ @DietrichBurde OP seems to be defining sign as the function $re^{i\theta}\mapsto e^{i\theta}$. Under that assumption, the sign of $i$ is $i$. OP, I've also seen sign used this way, but I've seen \sgn more often. $\endgroup$
    – Rushabh Mehta
    Commented Aug 11, 2020 at 20:37
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    $\begingroup$ @DietrichBurde: why this discussion about complex numbers ? The OP stays in the reals, doesn't he ? $\endgroup$
    – user65203
    Commented Aug 11, 2020 at 20:40

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The sign function is commonly written $\text{sgn}$.

https://en.wikipedia.org/wiki/Sign_function

What you discuss in your post seems to be related to the "sign transfer" operation,

$$\text{sgn(b)}|a|,$$ for which I don't think there is a usage (it is defined along with some programming languages).

A typical use of the sign function is in the expression of the numerically stable solution of the quadratic equation,

$$x_0=\frac{-b-\text{sgn}(b)\sqrt{b^2-4ac}}{2a},\\x_1=\frac{2c}{{-b-\text{sgn}(b)\sqrt{b^2-4ac}}}.$$

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