Wording in math

When it is said that there is "an equation in x" what exactly does that mean? It seems to me that it's saying that there is an equation inside of the variable x and it makes absolutely no sense to me because a variable usually contains a real or complex number, not an entire equation (though I am aware it could, but wouldn't the equation also have a variable in it?). So when I first read my Algebra textbook, I read the first sentence of the textbook "An equation in x is a statement..." I read that line over and over again trying to understand it because I wanted to gain an intuition for math but I just could not get it or understand it at all so I skipped it and just learned the rest of the material, but now I feel I am missing an entire picture of Algebra whenever I cannot understand the grammar of some textbooks.

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Originally Posted by daigo

When it is said that there is "an equation in x" what exactly does that mean? It seems to me that it's saying that there is an equation inside of the variable x and it makes absolutely no sense to me because a variable usually contains a real or complex number, not an entire equation (though I am aware it could, but wouldn't the equation also have a variable in it?). So when I first read my Algebra textbook, I read the first sentence of the textbook "An equation in x is a statement..." I read that line over and over again trying to understand it because I wanted to gain an intuition for math but I just could not get it or understand it at all so I skipped it and just learned the rest of the material, but now I feel I am missing an entire picture of Algebra whenever I cannot understand the grammar of some textbooks.

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To say that "an equation is in x" means that "the variable in the equation is x".

Well the latter sentence makes a lot more sense, so why don't authors just word it that way instead of seemingly intentionally confusing the readers? Especially for a beginner textbook when it is assumed the reader has no previous knowledge of Algebra, how are they supposed to figure out that that's what the writers mean?

You are expected to become fluent in the language of mathematics. Mathematicians like to have their own shorthands - this is why we have algebra in the first place...

Well I just think it would help if they defined the shorthands instead of just using it casually without explaining what it means

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Originally Posted by daigo

Well I just think it would help if they defined the shorthands instead of just using it casually without explaining what it means

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The phrase "An equation in x is a statement..." is exactly that — a definition. What comes before "is" is the defined term. The reader is supposed to disregard the commonsense meaning and associations of the defined term that he/she has; instead, a new precise meaning of the term is given after the word "is." In particular, you are not supposed to think that "in" here means "inside." Rather, the whole phrase "an equation in x" is a new indivisible term whose new meaning is given by the definition.

Mathematics recycles a lot of common words by giving them new meanings. We have definitions like "A group is a set with a binary operation satisfying the following axioms...," "A category is an ordered triple...," "A sequence of functions converges almost everywhere if...," "A meager set is a countable union..."

Dear daigo,
"An equation in x" simply means: "An equation in terms of variable x", i.e., "An equation involving variable x" or "An equation using variable x".
I hope it is clear to you now.

Neeraj Karn

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Originally Posted by emakarov

The phrase "An equation in x is a statement..." is exactly that — a definition. What comes before "is" is the defined term. The reader is supposed to disregard the commonsense meaning and associations of the defined term that he/she has; instead, a new precise meaning of the term is given after the word "is." In particular, you are not supposed to think that "in" here means "inside." Rather, the whole phrase "an equation in x" is a new indivisible term whose new meaning is given by the definition.

Mathematics recycles a lot of common words by giving them new meanings. We have definitions like "A group is a set with a binary operation satisfying the following axioms...," "A category is an ordered triple...," "A sequence of functions converges almost everywhere if...," "A meager set is a countable union..."

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It makes so much more sense when you're told to regard a word or phrase as non-English and instead, sort of a separate word or phrase in a different language. I've been trying to dissect these wordings grammatically but I didn't know I wasn't supposed to do that...I think it's kind of weird that the internet in general has helped me more in math than my professors or school textbooks.