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. $\endgroup$ – BPP Dec 15 '18 at 14:46
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    $\begingroup$ Another problem is language: For me the set of natural numbers $\mathbb{N}$ includes $0$ as natural numbers are cardinals of finite sets and $0$ is the cardinal of $\emptyset$; others call whole numbers the elements of $\mathbb{N}\setminus\{0\}$ but, in french and some other languages, "whole number" means "nombre entier" which can mean "entier natural" that is $\mathbb{N}\ni 0$ or more often "entier relatif" that is $\mathbb{Z}\ni 0$. $\endgroup$ – BPP Dec 15 '18 at 15:22
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    $\begingroup$ Even that standard says it is for natural sciences and technology, not for mathematics. $\endgroup$ – Gerald Edgar Dec 15 '18 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 '18 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. $\endgroup$ – James S. Cook Dec 16 '18 at 5:41


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 '18 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. $\endgroup$ – Daniel R. Collins Dec 15 '18 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 '18 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.


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.


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