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Is there a symbol used for an element which a function sends to zero?

Like this: $f(?) = 0$, where "$?$" would be the symbol for which I am wondering if exists.

I think this could be useful for instance in factoring polynomials: $ax^2+bx+c = a(x-?_1)(x-?_2)$ and the zeroes of functions are always interesting.

Frank Vel
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    In the case of polynomials, we call the zeros "roots". Hence it is sometimes written $a(x-r_1)(x-r_2)$. Although I would always specify first "Let $r_1,r_2$ be the roots of $f$ [...]". – Eff Dec 31 '14 at 13:09
  • @Eff That is true, but there is no general way of expressing such zeroes with a symbol? – Frank Vel Dec 31 '14 at 13:25
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    I don't know of a universal notation, which is why I didn't make an answer :-) – Eff Dec 31 '14 at 13:27

2 Answers2

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You could write $f^{-1}(0)$. But note that this is a set, not a single specific value, because there might be several different values that the function $f$ sends to zero.

Another notation you might see is $Z(f)$, standing for the "zero set" of $f$: $$ Z(f) = \{ x : f(x)=0\} $$

In some fields, like linear algebra, $Z(f)$ is called the "kernel" of $f$. See here.

Personally, I don't like using special symbols when words will serve instead. So, in my view, "$x$ is a zero of $f$" is much better than "$x \in Z(f)$". Of course, "$f(x)=0$" is pleasantly clear and concise, too.

bubba
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  • So there is no symbol for the element? I know about kernel, but the elements of the kernel has no special symbol, yet the set has? – Frank Vel Dec 31 '14 at 13:22
  • I don't know of one. If there is one, I'd guess that it's so obscure that few people will be familiar with it. Best to stick with well-knonw notations, to make yourself clear. – bubba Dec 31 '14 at 13:39
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In almost twenty years of mathematics I've never seen such a symbol. I believe that it would be pretty useless, since in many cases zero is nothing really special. It is mainly for convenience that we speak of zeroes of a function: everybody learns that $f(x)=y$ is equivalent to $f(x)-y=0$ (I am thinking of real- or -complex-valued functions here).

Everybody can introduce his own notation, but I guess you can hardly avoid a sentence like "Let $x_1$ be a zero of $f$". How can you distinguish different zeroes with a unique symbol?

Siminore
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  • Well if $?$ was a symbol for a zero, then $?_1$ could be a specific one? I Don't like the $?$-symbol though and was thinking maybe something existed... – Frank Vel Dec 31 '14 at 13:24
  • Wes, but why zero and not, let's say, one? Or two? – Siminore Dec 31 '14 at 13:43
  • I don't know any uses for setting a function equal to one or two, but I thought $0$ would be more useful. Although the set of the elements mapping to zero has a name, so I thought maybe the element itself had a name. I don't see any reason why not to extend this to other numbers though, but zero seems most fundamental. $0_f$ could be an element mapping to $0$, and $1_f$ an element mapping to one? – Frank Vel Dec 31 '14 at 15:18
  • Yes, of course you can use your own symbol. the issue here is that it is useful to speak about the set of all points such that... while it is rather useless to propose a symbol for the generic element of that set. Which one? Any? If you agree to use $0_f$, would you like to read a sentence like "Consider now $0_f$, and prove that it is larger than one"? – Siminore Dec 31 '14 at 15:46
  • The set could then be written $f^{-1}(0) = {[0_f]_1,[0_f]_2,[0_f]_3,\cdots,[0_f]_n,}$, I think it could be useful to name such elements. But sure, I wouldn't mind reading a sentence with $0_f$. – Frank Vel Dec 31 '14 at 15:58
  • Im my opinion, such a notation might be of some use only if $f^{-1}({0})$ was a finite set. If there was a whole interval of zeroes, it would be impossible to enumerate such points. Anyway, in mathematics a red hat is not necessarily red, and it is not necessarily a hat (cit.) We can't reserve a special symbol for everything that can be useful. – Siminore Dec 31 '14 at 16:22
  • Your suggestion that you “it would be pretty useless since in many cases zero is nothing really special” is utterly bizarre in light of the fact that large portions of algebra and calculus and algebraic geometry is precisely the study of the conditions under which various functions are zero-valued. – MJD Dec 31 '14 at 16:56
  • Well, and yet nobody felt the necessity to use special symbols for the zeroes of those functions. However I agree that my viewpoint is biased towards mathematical analysis, where, for instance, zeroes are less useful that fixed points. So I may wonder why we don't have a symbol for fixed points... – Siminore Dec 31 '14 at 18:59