A variable is a concept in mathematics that has an ability to vary across certain group of numbers under consideration.
An unknown quantity is a constant (number) that we need to solve using equation to find out the value.
It is customary to represent the unknown quantity by a letter (say x), which is generally called as variable, which is capable to take multiple values.
Why there is no need to differentiate between a variable (say $x$) and an unknown quantity represented by $x$?
Example:
It is fine to call $x$ as a variable in an expression $4x+3$. But $x$ in $4x+3=2$ is an unknown quantity, which is generally called as variable. In-fact it does not vary.
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$\begingroup$It is true that they are different. For instance, when learning to solve quadratic equations, you encounter$$ ax^2+bx+c=0 $$In this case, the letters $a,b,c$ ("blank boxes" for you to fill in whatever quadratic equation you happen to come across) logically mean something distinctly different from the letter $x$ (the unknown you want to solve for).
However, if you aim to manipulate the expression, they all follow the same algebraic / arithmetic rules. And some times, $a,b$ or $c$ can be the unknowns, so the letters switch roles. Thirdly, you want to be able to switch effortlessly between the above equation and the function $f(x)=ax^2+bx+c$, where the role of $x$ has drastically changed.
So being overly strict on exactly what a letter is may possibly have its merits in teaching, but in actual mathematics it often only confounds and hinders. Properly using the different names in writing may help your readers understand what you aim to show them. But it will not affect the actual math, and I will even go so far as to say that being able to "reinterpret" letters on the fly is an important skill (see, for instance, the second and third points in the previous paragraph).
$\endgroup$ 1 $\begingroup$Let me provide a computer science and in particular a type theoretical perspective on this question answering how both concepts can be unified in a formal sense. I am not entirely sure if those actually should be unified. The distinction between "things you know" and "unknowns" certainly does exist in mathematical thinking and writing. Perhaps our techniques for formalizing math, say in type theory, and for mathematical knowledge management in general should support making this distinction.
A variable is a concept in mathematics that has an ability to vary across certain group of numbers under consideration.
source: question by OP
Let me counter this viewpoint. Below you find two made-up sentences that both make use of "variables" in the commonly understood sense in the bold-faced parts with "let". However, I claim, that only one of them supports your viewpoint of variables denoting "varying" quantities.
"We want to prove $\forall n\in\mathbb{N}.\ n \text{ even} \Rightarrow \text{4 divides } n$. Hence, let $m \in \mathbb{N}$, then ... (introduction of a fresh undefined constant to prove a universally quantified statement ["forall introduction rule"]*)
*) via Curry-Howard this can also be seen as a binding of a function argument
"Let $V = \mathbb{R}^3$ abbreviate our vectorspace. Then consider the set of linear functions $V \to V$..." (introduction of defined constant $V$)
Indeed, the first case supports your viewpoint. There, $m$ can be thought of as varying over values in $\mathbb{N}$. However, in the second case, $V$ does not vary at all. It is fixed as an abbreviation. But would one commonly understand both sentences to use variables then? Let us unify both instances. We can see the first case as introducing a (locally-scoped, fresh) variable $m \in \mathbb{N}$ that is undefined; hence, whatever follows is true for all values. This corresponds to $m$ varying/ranging over all values. By contrast, we can understand the second case as introducing a (globally-scoped) variable $V = \mathbb{R}^3$ that is defined; hence, is fixed.
In fact, in many flavors of type theories, variables can either be defined or not and thus variables generally take the form $$c\colon A\,[=A]$$ where $A$ is some expression. Here, the part after the colon is the type, and the optional part after the equality sign is the definiens (also: definition). For instance, $m \in \mathbb{N}$ would be written as $m\colon\mathbb{N}$ in this way of doing things, and $V = \mathbb{R}$ would be written as $V\colon\text{vectorspace}=\mathbb{R}^3$.
An unknown quantity is a constant (number) that we need to solve using equation to find out the value.
source: question by OP
For me, an unknown quantity as in an equation is a proposition with some locally-scoped variables together with the task of finding instantiations, i.e. definitions, for those variables making the proposition true. Let us consider your example of finding $x$ such that $4x + 3 = 2$. This equation you can see as a context $\{x\colon\mathbb{N}\}$ of the only locally-scoped variable there is, the proposition $4x + 3 = 2$ in that context, and the question which instantiation of $x$, i.e. of the context, makes the proposition true.
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