Why do we write:
solve for x
solve for an/the x
Where t represents the number of tickets
Where the t represents the number of tickets
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In math and computer programing, variables and constants (such as
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.
If someone were inexperienced with math or perhaps if they were speaking to someone about mathematical notation, they might think of
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.
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
Β, or even
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:
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.
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.
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.
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:
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.