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If the mathematical concept of 'x', in simplest definition means 'any number', what would the opposite of 'x' be? What I mean is, what is the symbol, if there is one, to represent 'ALL numbers'? If there isn't one, why not?
Also, I'm not talking about infinity, nor sigma, but the concept of a symbol representing all numbers as a whole, including negatives, fractions, complex/imaginary, etc.
There is an interesting point behind your question, and it is broader than just the issue of numbers. The idea is that it is possible to negate a function or set, by complementation, but it is not possible to negate an individual. If x is a variable, then its job is to range over a domain of individuals, so it cannot be negated.
The logician Peter Geach used this fact to make an important point about the subject/predicate distinction in logic. If we see a sentence like "Alice is happy", how do we know which is the subject and which the predicate? In simple cases we might just observe that 'Alice' names a thing and 'happy' names a property, but in more complex cases, this explanation is not available. Geach's account of subjects and predicates is that a basic sentence can be thought of as a function satisfied by an individual. So "Alice is happy" is Happy(alice) in the same way we write a function as f(x). How do we know it is Happy(alice) and not Alice(happy)? Because if we negate the whole sentence to "it is not the case that Alice is happy" this means "Alice is not-happy"; it doesn't mean "not-Alice is happy". So we can negate (complement) 'happy' but not 'Alice' identifying it as the predicate.
What you ask for is a bit tricky because you are using an informal concept to describe variables, rather than the very exacting formal definition of what 'x' really means. In general, informal concepts tend to run into trouble when you try to take the opposite of them, because they weren't sufficiently rigorously defined to survive such a treatment.
That being said, there are many cases where we wish to talk about a domain of "everything." Such a thing is often called the "Universe," and it encapsulates every "thing" which one may wish to talk about using the language of mathematics. It is often given the symbol "U," but it is typically referred to using words first, to make its meaning perfectly clear.
This "Universe" does not behave like a normal number. It behaves more like a collection of things. In fact, the mathematical term for it is a "class" (which is in contrast to calling something a "set," which is a easier to understand collection).
From there, you can look at the "for-all" notation: ∀. I can write something like ∀x∈S, which is a series of symbols which means "for all possible values of x which are elements of the set S, the following expression is true." This for-all operator has its own behaviors in First Order Logic. You can also write something like ∀x, without the "element of S" specification. If this is done, it is assumed that x is any value within the domain of discourse (i.e. the Universe). This is probably as close to an "opposite of x" as you will get.
Of course, we also have several other meanings of the word "opposite." All of these are other concepts which are associated with the word "opposite," but have very different meanings than the one you appear to be referring to:
-x - additive negation of x1/x - multiplicative negation of x¬x - logical negation of x (strong)not x - modal negation of x (weak)x* - complex conjugate of xxᶜ - set theory complement of xIf, in fact, what you were actually after was a symbol for "all numbers" and the whole discussion of "opposites" and "x" was a false path, we do have symbols for that. Most of them are written in the so called "blackboard font":
You rarely see a symbol for "numbers" in general because mathematicians don't have much use for such a symbol. The different sets of numbers have different enough properties that mathematicians typically want to be specific as to which set they are referring to.
"x" is a place-holder. It has no meaning or effect in any way. It's what replaces "x" that has meaning. By itself, is is just an abstract idea.
no, because we cannot define "all numbers". Symbols like N for natural numbers work because we know (or can define) what natural numbers are.
A symbol like 'x' by itself does not stand for anything. In particular it does not mean "any number". It must be contextualized to something definite, e.g. "let x=3" or "let x be a member of N". Sometimes the context is not explicitly stated, but there always is one.
Suppose we were to propose that "A" is the symbol for all numbers. Then saying that x is a member of A would be vacuous. It would be a definition with no consequences, we would have no way if testing it.
If x is our concept, then !x would be it's opposite. Logically, the intersection of x and !x x ^ !x = {} should result in nothing (or an empty set). In your simple case of x = any number, then !x can't be all numbers, because then the intersection of x and !x would still be x, I'd say the opposite of any number would be no number, and the opposite of all numbers would be none of the numbers.
By x, you mean a single, arbitrary number. You are asking for a symbol to represent the whole class of numbers.
To represent multiple numbers, one possibility is to turn to set notation, typically represented by uppercase letters. For example, given S = {1,2}, S is a set that contains 2 numbers. By adding more elements, we can arrive at the concept of infinite sets.
You ask for "negatives, fractions, complex/imaginary, etc". There are standard mathematical symbols that cover the three classes specified, being the integers (ℤ), rationals (ℚ) and complex numbers (ℂ). (Fonts from Zah Lee.) For "etc", use standard set notation, e.g. S.
Your question appears to be asking for a convention for referring to all numbers. Note that ℂ includes the elements of the earlier two as subsets, but it's at least philosophically possible that there are even broader concepts of number. The convention I'm suggesting is set notation (which in turn has its own set of conventions).
set notation Sets are fundamental objects in mathematics. Intuitively, a set is merely a collection of elements or members. There are various conventions for textually denoting sets. In any particular situation, an author typically chooses from among these conventions depending on which properties of the set are most relevant to the immediate context or on which perspective is most useful. - wikipedia
