Why don't we use articles for variables in math problems?

Question

Why do we write:

solve for x

and not

solve for an/the x

or

Where t represents the number of tickets

and not

Where the t represents the number of tickets

We do sometimes include the article in contexts like where the n represents.... It's perfectly valid syntactically speaking - but as that linked chart shows, *it's not common.* So any "explanation" as to why we don't normally include the article would have to explain why we *sometimes* do. — FumbleFingers, Jul 13 '21 at 16:40
The difference in whether to use an article is in whether one is talking about the symbol or the value it represents. If you're talking about the symbol (i.e. "the letter n"), then there can (and usually are) many instances of "n"s in the world, so it is a generic term, which uses articles. If you are talking about the value represented by the symbol in that context (i.e. "n = 148.37"), then there is only one value, and it is named "n", so "n" is a proper noun (which does not use articles). — Foogod, Jul 13 '21 at 17:53
But we do sometimes use an indefinite article before a number in gaming contexts — primarily dice (I rolled a 4) and cards (I needed a 7 but I got a 2), but also sometime sports (he shot a 3). — Scott - Слава Україні, Jul 14 '21 at 16:23
Relevant questions I found over on [math.se], some with good links to follow: How to learn/speak “mathematical english”? ; the equation (A) vs Equation (A) ; Grammar Mistakes in Math Writing — AakashM, Jul 15 '21 at 09:27
more math grammar: How to best format the question mark that immediately follows an equation in a question sentence? — uhoh, Jul 15 '21 at 22:23

Ajedi32 · Answer 1 · 2021-07-13T17:16:08.313

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In math and computer programing, variables and constants (such as x and t in your examples) are treated like proper nouns (like names of people or places) rather than common nouns (names of a type of object). Just as you wouldn't say "I wonder what the Bob is doing today?" or "The Mary is coming over for dinner tonight." you also wouldn't talk about "the x" or "the y" in an equation.

I believe this is because in math, variables are used like names; they only ever refer to one particular "thing" in an equation. There's no need to make any sort of distinction about which x you are referring to. x is x.

If someone were inexperienced with math or perhaps if they were speaking to someone about mathematical notation, they might think of x or t as letters rather than as names and therefore use articles like "the" when referring to them (e.g. "the [letter] x in this equation refers to...") but that mentality is not common among those who work with equations on a regular basis.

edited Jul 13 '21 at 17:16

answered Jul 13 '21 at 16:50

Ajedi32

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This is a good answer, but I would actually probably phrase it more strongly: They are not just treated like proper nouns, that is inherently what they are. "x" is not a type of thing, it is the name which has been given to some (particular) value. There cannot be multiple different "x"s in the same context (with different values, etc), so saying "an x" or "the x" (which implies there can be more than one) is actually wrong, because it actually says something about "x" that isn't true. – Foogod Jul 13 '21 at 17:47
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@Foogod "There cannot be multiple different "x"s in the same context" Sure, there can be. Usually they're denoted by subscripts. For example, you might sum x1, x2, x3, ... xn. – nick012000 Jul 14 '21 at 01:44
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There might be borderline cases such as "In $\pi(n)\sim \frac {n+\sin(\pi n)}{\ln n}$, note that the $\pi$ on the left stands for the prime counting function and the $\pi$ on the right for the well-known constant". But this may happen as well in "real life" when distinguishing between several people with same proper name. – Hagen von Eitzen Jul 14 '21 at 04:56
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@nick012000 in that case though, each object still has a unique name. In that context there may be multiple x's, and one might refer to one of those with an article, but each object x_1, x_2... is still unique, and cannot take an article – Tristan Jul 14 '21 at 09:35
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In addition to being a name, "x" is shorthand for "an unknown quantity" or "the unknown quantity", etc. So, saying: "Solve for the x" would be akin to saying "Solve for the the unknown quantity" or "Solve for the an unknown quantity." But, going beyond this example ("Solve for x") thinking of any letter as a name is probably more appropriate (e.g. a name given to a specific quantity). – TravisJ Jul 14 '21 at 12:59
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@nick012000 In addition to what Tristan said, you do use an article when you refer to them as a group: "the x's". Similar to "Trump was President" versus "The Trumps are a rich family" – Barmar Jul 14 '21 at 14:43
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@nick012000, what about quadradic equations? X takes on multiple values, and it would be common to talk about the positive X or the negative X. There are plenty of equations that likewise yield multiple values for a variable as well. Coming from a math heavy field myself, it isn't a common occurence, but an exception nonetheless. Collectively they carry the same "weight" as a name, but individual values would be distinguished by an article, e.g. "The X that is most reasonable in this context is..." – Ryan Jul 14 '21 at 16:14
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This mathematician writes "the variable x" not infrequently. It's a useful device to avoid starting a sentence with a mathematical expression, to differentiate variables from constants, or even to give variables types, like "the vector v." Writing math problems, I often even write something like "find all values for the variable x that satisfy the following equation" rather than "solve for x." – Kyle Miller Jul 14 '21 at 18:25
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@Ryan: I have never heard "the positive x" or "the negative x." I was taught to call them "the positive/negative zero/root/value/solution" instead. But never "x." – Kevin Jul 14 '21 at 20:30
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This doesn't really answer the question. Some proper nouns are preceded with "the" and some aren't. For example, the Mississippi River, the Oregon Trail, the NSA, and the Amazon. – David Schwartz Jul 14 '21 at 22:18
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@DavidSchwartz: It does answer the question. The fact that there are exceptions to the rule is grounds for a different question on why that's true, not evidence that citing the rule is a non-answer. Note that in all your examples, we're referring to "the [specific] [object type]", with "the Amazon" dropping the implied "rainforest" object in practice. We might likewise say "the Bob person is different from the Alice person" if we're focusing on people as objects. – MichaelS Jul 14 '21 at 23:42
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It might be useful to point out the exception as it applies to math that's been alluded to in the comments. The main exception is if you have multiple equations that use the same symbol for different values. In "A + Bi", 'i' likely refers to √(-1), where in "Σ i=1 to 100 f(x)", 'i' is an index variable, and in "V = IR", 'I' refers to current. All three could conceivably be used in the same math in electrical engineering, so you might say things like "this i" or "the complex i" to be clear. (Though you're more likely to see the first 'i' become 'j' and the second become 'n' for this reason.) – MichaelS Jul 14 '21 at 23:53
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@HagenvonEitzen — And to your point, you used the phrase "the π" a couple times in your comment to distinguish between the differing usages :-) – Jivan Pal Jul 15 '21 at 01:44
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Articles like "the pi" or "an x" are used to refer to occurrences of variables or constants within formulae, rather than to refer to the variables or constants themselves. Consider the sentence: "this fraction has an x on the top and another x on the bottom, so we can simplify it as long as x isn't zero". In this sentence, "an x" and "another x" could be written more formally "an occurrence of x" and "another occurrence of x", but the clause "x isn't zero" refers to the variable x itself rather than an individual occurrence of it within a formula, so no article is used. – kaya3 Jul 15 '21 at 11:54
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Alternatively, "the pi" can refer to an occurrence of the symbol pi, as opposed to an occurrence of a variable or constant (or function) whose name is pi. In the sentence "the pi on the left is the prime counting function, while the pi on the right is the circle constant", each "the pi" refers to an occurrence of the symbol pi, but despite being occurrences of the same symbol, they are not occurrences of the same variable, constant, or function. – kaya3 Jul 15 '21 at 11:59
@MichaelS Surely when we say "X" in math, we could equally well say "the X variable". And we could also omit the word "variable". These are all perfectly valid English choices and the question is why math makes the choice to simply say "X". All three practices are very common in English. If there's a reason mathematicians use "X" instead of "the X variable" (or "[find] the X value" or "the value of the X variable" as in "the Amazon rainforest") or "the X" (as in "the Arctic"), this answer doesn't explain it. – David Schwartz Jul 15 '21 at 14:58
@DavidSchwartz I wouldn't say "the x variable" under any circumstances I can imagine (other than in a direct quotation, as here). It's "the variable x". – David K Jul 15 '21 at 17:20
@DavidK I agree. And both forms are used with proper nouns in English. I'd say, "the elephant Dumbo" but "the Amazon rainforest" and "the river Styx", but "the Valsalva maneuver". – David Schwartz Jul 15 '21 at 17:28
@DavidSchwartz https://english.stackexchange.com/questions/59271/why-there-is-the-before-some-names-but-not-others – user3067860 Jul 15 '21 at 20:17
@DavidSchwartz If the point is that the usage is not simply based on the premise that $x$ is a proper noun, the point is well taken. – David K Jul 15 '21 at 22:41
In the context of computer programming, I might say "In the command x := x+1, the x on the left refers to a location but the x on the right refers to a value." – Andreas Blass Jul 16 '21 at 17:59
The I disagrees with the fundamental premise of this answer. – John Smith Jul 24 '21 at 07:08

Austin Hemmelgarn · Answer 2 · 2021-07-14T15:27:53.463

It’s a matter of context, and to a lesser extent jargon.

A variable in mathematics actually has nothing to do with the symbol or name used to identify it. x² + y³ = 0 will always have the same solutions regardless of whether the variables are identified as x and y, Α and Β, or even ζ and Д. However, in any given form of an equation, the symbol or name used to identify a variable is unique, and therefore functions as a proper name.

In English, with limited exceptions, you only use an article with a proper name for one of four reasons:

The proper name is being used in the manner of a descriptor to disambiguate something. For example, in ‘The City of Kettering’, the article is attached to ‘City’, and the proper name ‘Kettering’ is being used to identify which city it is.
The proper name inherently includes the article. This is relatively rare outside of nicknames (for example, Donald Trump is sometimes known as ‘The Donald’), but does occur in other cases. For example, the rather famous nearly symmetrical mountain peak in the Alps is almost invariably known in English as ‘The Matterhorn’, and the large country covering most of the southern half of North America is invariably ˘The United States’ (clipped from ‘The United States of America’, which uses an article for the first reason above).
The speaker or writer wishes to emphasize the identity or authenticity of the thing being spoken of. In most such cases in spoken English, there will be clearly audible verbal emphasis on ‘The’, and in many such cases in written English, the word ‘The’ will be italicized to indicate emphasis.
The proper name is being used in a generic or categorical sense, instead of as an identifier for a specific entity. For example ‘I am an Austin.’ or ‘She looks like a Susan.’.

Usage of variable names in mathematics doesn’t fit any of these cases, so articles just aren’t used in a vast majority of cases.

It doesn't really even matter if it is unique, in terms of being a proper name--see the comments on the other answer discussing when multiple quantities that are all commonly represented by the same symbol, and sometimes those collide in the same equation. But if I know two people named John, John is still a proper name. So I work with John and I am friends with another John, and I ate lunch with the John that I work with. (Mathematically that's a problem, but not at all a problem for English.) — user3067860, Jul 15 '21 at 20:54

Davislor · Answer 3 · 2021-07-15T03:30:36.220

I sometimes hear articles in front of a variable name, if it’s to distinguish which of several possible assignments we mean. For example, if we have a sequence whose elements are a-sub-0, a-sub-1, and so on, I might specify “The a-sub-i minimizing the difference between a-sub-i squared and v,” or “an a-sub-j satisfying the inequality, ....” Afterwards, I would normally not use an article in front of the variable name, since it needs no further qualification.

You still do not need articles in either sentence, however.

score 3 · Answer 4 · answered Jul 15 '21 at 21:22

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Nouns are often described as being a person, place, or thing, but that's not quite correct. Common nouns actually are category labels. For instance, the word "book" labels the category of books. In English, simply referring to "book" is not grammatical, because "book" does not refer to a thing but a category. We need a determiner to refer to an actual book.

Variables, on the other hand, do refer to a particular thing (usually a number), rather than a category; in that sense, they act like proper nouns.

answered Jul 15 '21 at 21:22

Acccumulation

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"I enjoy reading books" seems perfectly grammatical to me. – Phil Frost Jul 16 '21 at 15:02
@PhilFrost Yes, but "I enjoy reading book" is ungrammatical. – Acccumulation Jul 16 '21 at 17:43

Phil Frost · Answer 5 · 2021-07-16T15:14:59.810

I believe the reason can be demonstrated with an example:

Solve 7x² + 13x = 9 for the x.

Which one? The first x or the second x?

Solve 7x² + 13x = 9 for an x.

So I can just pick either one? Here you go: x = (9-7x²)/13

This is of course not the solution most people are expecting.

The point being, there are two things here:

A quantity, which we have named "x".
Some "x" symbols on the page which represent that quantity.

When you use an article like "the" or "an", you indicate you're talking about a specific "x" symbol, not the quantity named "x".

Likewise, your favorite food is not "a banana" or "the banana". You would say, "my favorite food is bananas", because your favorite food is the concept of bananas, not a specific banana. But you would say "please pass me a banana" or "I'll have the banana on the left" because these sentences are talking about a specific banana.

The second one could be useful as a fixed-point iteration that converges to one of the two possible solutions. Still, it's not what's usually meant by “Solve for x.” — dan04, Jul 16 '21 at 16:53

Why don't we use articles for variables in math problems?

5 Answers5