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[To the best of my knowledge, I did not find any previous question that deals with this issue. I hope it is not a duplicate.]

I have a problem with the way in which mathematicians sometime use words instead of numbers to actually denote numbers. To me it seems that numbers should always be used whenever we are actually talking about numbers, however it is not really the case.

EXAMPLES:
1) "Probability one" or "Probability 1"?
2) "Measure zero" or "Measure 0"?
3) "Solution larger than one" or "Solution larger than 1"?
4) "The value is one" or "the value is 1"?

In all these cases, I think the proper way of writing is the second one, however I often find the first way.

To me this looks particularly incoherent, because the same authors shamelessly move to the second style whenever the number is slightly awkward (i.e. it is not an integer). Indeed, it is quite easy to find somebody who writes down "probability one", and then later on writes "the value is $1/2$". [Please, note that I am not referring to a sentence like "the value of $\alpha$ is $1/2$", that can be easily rephrased in a more readable expression such as "where $\alpha = 1/2$".]

Here there are my questions.
1) Is there a general rule to write down numbers?
2) Is my intuition correct about the way in which things are usually written?
3) Is my intuition correct about the way in which things should be (on logical and grammatical grounds) written?

Thank you in advance for any feedback!

PS: True enough, the first and the second examples can be considered slightly different from the third and the fourth, because - even if it actually applies what I claim (i.e. writers use words to denote numbers) - those could be considered expressions that have developed historically an inner meaning that goes beyond the fact that they indirectly refer to numbers.

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  • $\begingroup$ Be glad we are not required to say "unity", as mathematicians used to do when writing the number $1$ as a word. $\endgroup$
    – GEdgar
    Aug 26, 2014 at 14:25
  • $\begingroup$ In german I learned that only the integers $0,...,12$ should be written out as words, everything else should be written as number. But this is rather a general rule, an does even in german not necessarly apply to mathematical texts. $\endgroup$
    – flawr
    Aug 26, 2014 at 14:25
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    $\begingroup$ I think this has less to do about mathematics and more to do about writing style. When I was going through elementary and secondary school, the rule of thumb was to use words for numerals less than 10, and to use the numerals for 10 and up. $\endgroup$
    – Hayden
    Aug 26, 2014 at 14:25
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    $\begingroup$ @Hayden (and flawr): Yes, non-mathematical writing has general rules like this. Kolmin says it is illogical for mathematical writing. I say to Kolmin: go ahead and to it your way, but don't complain when others do it in other ways, $\endgroup$
    – GEdgar
    Aug 26, 2014 at 14:28
  • $\begingroup$ @GEdgar: I see what you meant. Thus my question is, which one of the two forms I presented do you consider good mathematical writing? (More or less in the same way in which there is bad mathematical writing for Serre). $\endgroup$
    – Kolmin
    Aug 26, 2014 at 14:31

1 Answer 1

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I found a guide on writing mathematical papers here (about in the middle of the text) that has following guidlines:

(...) In addition, avoid numerals because they slow down the reading. Write numbers out if they can be expressed in one or two words and are used as adjectives, unless they are accompanied by units, a percentage sign, or a monetary sign. For instance, write, "The equation has two roots," and "One root is 2." Don't begin a sentence with a numeral or a symbol; reformulate the sentence if necessary.

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    $\begingroup$ Thus, the second way should be the best one, in terms of mathematical style as well, shouldn't be? Actually, it is a bit ambiguous, because in one line it says that the have to (note the imperative) be written out if they can be expressed in one or two words, but then we find "One root is 2". Now, this looks good to me, but it is slightly incoherent, isn't it? $\endgroup$
    – Kolmin
    Aug 26, 2014 at 14:41
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    $\begingroup$ The remark that "numerals slow down reading" is ridiculous. Generally, using appropriate mathematical notation (vs. natural language) speeds-up reading and comprehension (assuming a mathematically competent reader). I'd dread reading a paper in number theory written that way. $\endgroup$
    – Bill Dubuque
    Aug 26, 2014 at 14:41
  • $\begingroup$ IMO, both yours and the site statement are debatable. It is true that mathematical writing is more about conventions than logical soundness, but still using numbers does not interfere with the quality of the reading and it is generally sounder (at least in the examples I wrote down). $\endgroup$
    – Kolmin
    Aug 26, 2014 at 14:45
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    $\begingroup$ I do not fully agree with @BillDubuque's comment. @ Kolmin: Yes it is coherent, since the author said you should write it out when the number if used as an adjective. The way I understand it is that they can only be written out when used as adjective. But I agree, I prefer numerals when refering to a value. (But not as adjective) $\endgroup$
    – flawr
    Aug 26, 2014 at 14:46
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    $\begingroup$ @Kolmin Further, natural language may inhibit intrinsic mathematical pattern matching. For a simple example, compare: "in a domain, $\, x^{\color{#c00}{\large 5}}-a\,$ has at most $\,\color{#c00}5\,$ roots", vs. "five roots". Using precisely the same symbol $\,\color{#c00}5\,$ in both places helps the reader to see the relationship between the degree and the number of roots. In less trivial examples, such pattern-matching can be greatly obfuscated by the use of natural language. $\endgroup$
    – Bill Dubuque
    Aug 26, 2014 at 15:18

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