ISO 80000-2:2009 is a standard describing mathematical signs and symbols developed by the International Organization for Standardization (ISO). In teaching mathematics, should one always follow this standard?

As an example, there is no universal agreement on whether zero should be considered as a natural number or not. But since there is an international standard which considers zero as a natural number (though the teacher may not like this convention), should one teach base on this standard only in order to avoid any confusion?

Another example is, people use $\log x$ for both natural and common logarithm. In my teaching I usually spend 10 minutes explaining the difference and the fact that in most calculators, $\log$ means common logarithm and in WolframAlpha, $\log$ means natural logarithm. My personal rule of thumb is, $\log$ means natural logarithm starting at pre-calculus and means common logarithm before pre-calculus. Based on ISO 80000-2, one should use $\lg$ for common logarithm and $\ln$ for natural logarithm. Under this convention, ambiguity ceased to exist.

Edit: The Chinese goverment published a standard in 1993, requiring all institutions in China teach that $0$ is a natural number. So at least in China, the dispute is forcefully solved.

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    $\begingroup$ You should follow your country standards. The mathematical notations and terminology never were universal and probably never will be. $\endgroup$
    – user5402
    Dec 15, 2018 at 14:46
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    $\begingroup$ Even that standard says it is for natural sciences and technology, not for mathematics. $\endgroup$ Dec 15, 2018 at 16:43
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    $\begingroup$ The nice thing about standards is that there are so many to choose from. -- Andrew S. Tanenbaum $\endgroup$
    – Adam
    Dec 15, 2018 at 18:03
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    $\begingroup$ @Zuriel yes, learning that part of math is merely convention is part of math. It's the part of math which is not black and white. At least, it's part of the grey in math, I'm not so sure it's all of the grey. $\endgroup$ Dec 16, 2018 at 5:41
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    $\begingroup$ @Adam That quote reminds me of this xkcd. $\endgroup$
    – user5402
    Dec 21, 2018 at 14:03

5 Answers 5



This standard may be useful for professionals in international settings.

Most teaching happens in smaller, localized settings and things will differ from country to country (e.g. how large numbers are packed in bunches of three or how decimal places are separated from the integral part).

Focusing on a single standard will make it harder for students to adapt to different nomenclature that they will encounter at some time.

As an example, in the ISO standard only $\varnothing$ is listed as "empty set". Not everyone agrees on this and I guess that $\{\}$ is just as common. Your students will have a hard time to deal with this different notation if they are taught that there is the one way™ to write things down.

Consistency is important, but don't overdo it.

  • $\begingroup$ I do think there needs to be a place for everything. And you can't solve every issue in every question. Still. I find that there is this...trend...of wanting to discuss or advance very nuanced issues of math itself (sometimes even crankish, but in any case very picayune). Versus the key issues of concern to educators: methods of pedagogy, class discipline, how to get a job, how to be self actualized in career. NOT base 12 or log versus ln. $\endgroup$
    – guest
    Dec 15, 2018 at 19:45
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    $\begingroup$ I don't see how having $\varnothing$ be the dedicated symbol for "empty set" precludes the expression $\{\}$ from also meaning the empty set... it's just that it's not a single symbol. Likewise, $\{1\}\cap\{2\}$ and $\{3\}-\{3\}$ each indicate the empty set without being a single symbol, and leaving them out of some definition does not contradict their implicit meaning. $\endgroup$ Dec 15, 2018 at 20:27
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    $\begingroup$ Nitpicky teachers might deduct points for wrong symbols ({}) or not completely simplified expressions ({3}\{3}) when the chose to follow the standard strictly. $\endgroup$
    – Jasper
    Dec 16, 2018 at 8:46

The standard that you link to (ISO 80000-2:2009) seems to be not available for free. That is, in order for me to follow the standard, I have to be able to read it, and in order for me to read it, I have to pay for it. (It currently costs 158 Swiss francs or approximately 158 US dollars.)

To require mathematics educators to follow the standard implies requiring each to pay a relatively large amount of money.

It seems to me that if you want to require mathematics educators to follow the standard then you should make the standard freely available.

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    $\begingroup$ I've written mark schemes that award credit for notation compliant with the parts of ISO 80000-1 and ISO 80000-2 reproduced in the free-of-charge SI brochure, but I'd consider it unethical to do the same for the whole of ISO 80000-2, for this reason. $\endgroup$ Nov 25, 2020 at 15:59
  • $\begingroup$ @DanielHatton, thanks for the link. $\endgroup$
    – JRN
    Nov 26, 2020 at 1:58

Another point. It seems to me that someone who's trained to think that there's only one way to write everything might well be trained to think less in general. If you know that $\mathbb N$ may or may not include $0$, you're forced to check which definition is meant, and the awareness that it's important to define the things you use is reinforced. Later on, awareness of multiple conventions might lead to a greater willingness to define your own notation in situations where that's necessary or helpful.

Someone who thinks the whole of mathematics has been handed down "as is", including the set of symbols and operations and functions, might be less able to think flexibly and maybe, less able to make the contribution they might have been capable of.


I suggest following the convention in your country or institution while mentioning others. ISO 80000 is not a widely accepted standard in every fields. Many mathematicians strongly disagree with some of the decisions of the ISO standard (See section 4 of https://tug.org/TUGboat/tb41-1/tb127gregorio-math.pdf). In practice, publications from AMS and ACM do not follow the ISO standard as well.

The Chinese standard you mentioned was a mandatory standard which was written based on ISO 31-11, the predecessor of ISO 80000. It was turned to a recommendatory standard in 2017.


In particular, $\mathbb{N}$ is used to mean either $\{0, 1, \dotsc\}$ or $\{1, 2, \dotsc\}$, depending on the exact area/author. Some write $\mathbf{N}$ for this. It is important for your students to know there are several meanings in use for this particular notation, even if you pick one and stick to it religiously. Personally I stick to the second (most often used), and write $\mathbb{N}_0$ for the first when needed.

More generally, for some notations there are several conflicting definitions, while some notions have a lot of different notations (e.g. $f(x) = O(g(x))$ or $f(x) \ll g(x)$).

Always check any (perhaps just local) definitions. Mathematics notation and terminology is a living, changing language, with it's own dialects and local quirks.


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