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My instinct is often that there should be some technical word for the common class of equations and inequalities. That is: statements that connect two numerical expressions, with a relational operator in the middle. Granted that "relation" collects the operators of equality, lesser-than, etc., it seems like there should likewise be a common word for the class of statements using those symbols.

One halfway common name I see is mathematical sentence. I'm not sure how widespread this really is; and it seems to lack the feeling of a technical mathematical term. In fact, today on social media I saw a number of parents of school children react vehemently to the phrase, saying that it had to be obviously nonsense. (Here's a more technical definition that I doubt they'd be any happier about.)

So: Is there any more widely-used convention in mathematical circles for a name of the common class of equations and inequalities? Or is "mathematical sentence" the best we have?

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    $\begingroup$ What grade level are we talking about? If we're talking 3rd year in college, learning to use the precise mathematical terms may be desirable. If we're talking 1st graders, the emotional content of the term will be more important. $\endgroup$ – Cort Ammon Oct 7 '20 at 16:56
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    $\begingroup$ I don't know of a reasonably commonly used term, or a term that is sufficiently descriptive that it doesn't matter whether it's commonly used, that can be used for students at or below the elementary calculus level. This lack has sometimes caused me to try to find or make up (according to which of the previously described cases holds) such a term, but I don't believe I've ever found one, and instead I just "wrote around it". So I'm interested in what others might suggest. The "mathematical sentence" term might have been OK back in the 1960s and early 1970s new math era, but probably not now. $\endgroup$ – Dave L Renfro Oct 7 '20 at 17:00
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    $\begingroup$ also, would you put "x<3 ∧ y=4" in the class of sentences you are interested in? What about isSuccessor(3, 4), which is true for the usual definition of the "successor", but does not use one of the traditional ordering operators. $\endgroup$ – Cort Ammon Oct 7 '20 at 17:04
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    $\begingroup$ If you are posting an answer to the question, please post it in the answers to the question, not the comments. Mini-answers in comments prevent voting, prevent marking the answers as accepted, prevent coherent comment discussions, and inhibit others from posting answers. $\endgroup$ – Chris Cunningham Oct 7 '20 at 18:04
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    $\begingroup$ I agree with Amy B that "mathematical sentence" is fine. I suppose it's possible to find reasons against using it, but parental outrage related to math on social media is about the worst one I can think of! I considered an answer suggesting "equineqs" (pronounced eek-win-eeks) but I thought people would (wrongly) think I was joking. $\endgroup$ – Thierry Oct 11 '20 at 15:11
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My phrase has always been math "statement". Equations and inequalities clearly assert/state a relationship between two or more things.

My go-to direction using this would be something like: "Clearly explain the meaning of each math statement below in the context of the problem you just read." Then I list things like $f(4)=3$ or $g(7)>g(1)$.

I still use the word "expressions" when there is no stated relationship between two things, such as $7$ or $\gcd(8,12)$ or $4==2$.

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  • $\begingroup$ I especially like this answer, because you can use analogies to drive home the difference between expressions and equations: "Is 'Today is Monday' true?" and "Is '5=5' true?" vs "Is 'Monday' true?" and "Is '5' true?" $\endgroup$ – CosmoVibe Oct 15 '20 at 18:03
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I'd go with "simple relational sentence."

  • "sentence" - as you have researched, this is the technical phrase for a well formed formula with no free variables. What you describe here fits that bill
  • "relational" - The particular kind of operator you are looking at is a relational operator. One might be more pedantic and call it an "ordering" operator, but I think there's a better intuitive sense that comes from "relational." The idea of ordering only comes when you consider sets of values and determine if they are ordered or not.
  • "simple" - Presuming you want to exclude sentences like "x<3 ∧ y=4" which have a conjunction in them, you'll want to add an adjective like "simple." I like "simple" because its typically not a well defined math term. Simple is always relative. This gives you some wiggle room as to why your definition might not line up with anyone else's. "Atomic" would probably be more precice, but I think that precision may harm rather than help.

I think that the qualifications on "sentence" do a good job of avoiding the concerns parents might have with "mathematical sentence." I would argue that "Three is less than four" is also a simple relational sentence, and I think that gives credence to the name in the minds of anyone who wants to think of "sentences" as a purely English concept (or other natural language).

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I taught 1st through 5th graders from 1990 - 2015 and we used the term: number sentence.
Students were very comfortable with this phrase.

Added on Oct 15, 2020: I am currently writing questions for 4th grade achievement tests. I just wrote a multiple choice question asking, "Which inequality is true?" and my editor asked me to revise it as "Which number sentence is true?" It would seem number sentences are still preferred and used!

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    $\begingroup$ I did some googling a day or two after I wrote The "mathematical sentence" term might have been OK back in the 1960s and early 1970s new math era, but probably not now at the end of a comment to the question, and apparently "sentence" is still used in school mathematics (especially algebra 1 and below). In thinking about why the word "sentence" is used instead of the word "statement", the latter of which is pretty much universal once you get to precalculus and beyond, I wondered if perhaps "sentence" is a more age-appropriate vocabulary word than "statement"? $\endgroup$ – Dave L Renfro Oct 11 '20 at 14:31
  • $\begingroup$ FYI, "sentence" was used when I was at this level (grade school math to algebra 1), but that was during the mid 1960s to early 1970s (i.e. mostly within the new math era). $\endgroup$ – Dave L Renfro Oct 11 '20 at 14:33
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    $\begingroup$ @DaveLRenfro As you can see I modified what I wrote to clarify that I taught through 2015 and we continued to use number sentence (not mathematical sentence). $\endgroup$ – Amy B Oct 11 '20 at 15:41
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    $\begingroup$ I didn't even notice that you used "number" instead of "mathematical"! The phrase "number sentence" sounds so strange to me, but nonetheless it might have been used in my school mathematics. (moments later) I just checked my algebra 1 text (Dolciani), and in Section 2-4 it says: An equation, such as "$x+4=6,$" or an inequality, such as "$y-1 > 5,$" which contains one or more variables is called an open sentence. (italics not in original; phrase included in the book's glossary at the back) Also, on p. 1 the equality $3 + 1 = 4$ is called a statement (and "statement" is italicized). $\endgroup$ – Dave L Renfro Oct 11 '20 at 17:47
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    $\begingroup$ @NickC 4th graders do use variables in one step equations: A + 37 = 100. The inequalities are strictly numerical - they are not ready for variables inequality. $\endgroup$ – Amy B Oct 15 '20 at 19:56
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Is there any more widely-used convention in mathematical circles for a name of the common class of equations and inequalities?

Well, following the Dictionary of algebra, arithmetic, and trigonometry by S. Krantz, we can say that the equations (assertions of equality between mathematical expressions...) and inequalities (expressions that involve members of some partially ordered set...) are in the class of the expressions. However, it seems this class is bigger than you want.

expression A mathematical statement, using mathematical quantities such as scalars, variables, parameters, functions, and sets, as well as relational and logical operators such as equality, conjunction, existence, union, etc. (Krantz Dictionary)

In order to be more specific, you could use relational expression (as defined in Vivaldi's book).

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    $\begingroup$ Huh. I respect Krantz's work a lot, but this surprises me. I'm used to "expression" universally being used for a fragment that lacks a relational operator. Those books both look very interesting, thanks for that. $\endgroup$ – Daniel R. Collins Oct 9 '20 at 14:14
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Simplifying Cort's answer, what about relation? This fits the lay meaning of the word, and the definition of relation as an extension of the concept of function.

Edit In response to Daniel's comment: I agree that relation falls short of being the correct term. What is lacking is a modifier: What about "A relation of the form $=$ or $\leq$ is called a numerical/quantal/quantitative relation"?

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  • $\begingroup$ I think "relation" connotes a larger space than I'm talking about. I want to a term about specific pieces of writing, not subsets of Cartesian products in the abstract (which can be communicated in various forms,e.g., set notation, a table, a formula, etc.) $\endgroup$ – Daniel R. Collins Oct 8 '20 at 1:26

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