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This might be a silly question, but I'm curious if there is some sort of test to see if a number is positive or negative. What I mean by a test is that there is something that can be computed. For example, one test you might do to check to see if a complex number is real is that if $\overline{z}=z$, then we can conclude that $z\in\mathbb{R}$. Such a property like this is fairly useful and can be used to show that for example the eigenvalues of a self-adjoint operator are real. Furthermore, the complex conjugate is "nice" in the sense that $\overline{z\cdot w}=\overline{z}\cdot\overline{w}$ and $\overline{z+w}=\overline{z}+\overline{w}$, so if I have an algebraic expression that I am working with and wish to show is real, I can just apply the complex conjugation to each term individually.

This suggests the test that $x\in\mathbb{R}$ is positive if $\vert x\vert= x$. However, this "test" seems a little more unsettling to me for the reason that $\vert\cdot\vert$ does not have the same algebraic properties that complex conjugation enjoys (it is multiplicative, but we can't separate addition the best we can do is use the triangle inequality which doesn't seem helpful if we are looking at some large algebraic expression). Thus, I am curious if there might be a different test that might be a bit "nicer" in that it has more exploitable algebraic properties.

Edit: Perhaps to add some more context, I am not working with explicit numbers as it is pretty clear that if I have say $-2$ it is negative as compared with $6$ it will be positive. What I am more interested in is if there is some algebraic expression for example if we have some exponential sum like a Ramanujan sum

$$ c_q(n)=\sum_{\substack{1\leq a\leq q\\ (a,q)=1}}e^{2\pi i\frac{an}{q}} $$

Then using the complex conjugation trick shows that $c_q(n)$ must always be real; however, the sign can be either positive or negative. Similarly, if I have some convergent series $$ \sum_{n=1}^\infty a_n=L $$ where the $a_n\in\mathbb{R}$, could there be some sort of test to see if $L$ is postive or negative without knowing the value of $L$. That is can there be some kind of algebraic trick or test or algorithm that is nicer than taking the absolute value?

Steven Creech
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    If you are referring to $\text{sgn}(x)=\begin{cases}1 & x>0\ -1 & x<0\ 0&x=0\end{cases}$, then yes I am aware, but my question is more referring to if I have a general algebraic expression is there a "nice" way to compute the sign function. – Steven Creech Aug 22 '23 at 16:47
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    If $x=\sqrt {x^2}$ then $x \ge 0$ – Henry Aug 22 '23 at 16:50
  • $\Bbb R$ is a subfield of $\Bbb C$, so one can expect “nicer algebraic properties” when testing for real numbers than for for positive numbers. – Martin R Aug 22 '23 at 16:54
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    In what situation would you "test" your number? Usually a number comes with it's sign? or you can teat by multiplying with itself, it keeps the sign if is positiv, it changes if it is negativ. and your test for real komplex numbers how will you really do it? you first have to have the the conjugate number? – trula Aug 22 '23 at 16:54
  • From an axiomatic perspective, the sign function is (nearly) one of the primitives of the real numbers, similar to the additive inverse function. Would you have the same question about computing the additive inverse? – Lee Mosher Aug 22 '23 at 16:54
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    $\text{sgn}(x)=\frac{\sqrt{x^2}}x=\frac x{\sqrt{x^2}},x\ne 0$ – Тyma Gaidash Aug 22 '23 at 17:05
  • $sgn(z) = e^{i \arg z}$ – vvg Aug 22 '23 at 17:40
  • I don't find your edit particularly helpful. "Nice" is in the eye of the beholder. For example I, myself, would consider the axiomatic structures of the real numbers to be the "nicest" structures (e.g. addition, multiplication, identities, inverses, inequality, supremum) from which all other structures (e.g. the square root function on $[0,\infty)$) are derived by more and more complicated and messy formulas. – Lee Mosher Aug 22 '23 at 17:53
  • In general , it is impossible to decide whether an expression (including limits , sums , etc.) , is $0$. Hence , I guess that the sign cannot be determined in general either. – Peter Aug 22 '23 at 17:53
  • Furthermore, since the field axioms (addition, mulitplication, identities, inverses) are independent of the order axioms, there certainly is not any formula involving only the field operations which will yield positivity/negativity. – Lee Mosher Aug 22 '23 at 17:58

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