The concept of 'unknowns' in equations

[Jehannum]

You can solve an equation with one unknown. You need two different equations for two unknowns, and so on.

This implies a binary distinction between 'known' and 'unknown'. But are there grades or shades of unknown? For instance, if you know that a variable is negative but not its magnitude, can that help solve an equation? If so, is there any field of study on what the different degrees of unknownness are?

 

[mfb]

For instance, if you know that a variable is negative but not its magnitude, can that help solve an equation?

Sometimes, e. g. you can find a unique solution for $x^2 = 4$ if you know x is negative.
You can express this as an inequality, x<0, or directly specify the domain: $x \in (-\infty,0) \subset \mathbb{R}$.

As long as you don't go to probabilistic analyses, all that knowledge can be expressed in one of the ways described above.

 

[QuantumQuest]

Talking in the context of a specific problem in basic math, an entity or quantity is either known or unknown. In order to be labeled as known, it must fulfill some criteria or in other words, some properties of it must be precisely known, one of which is its value. If on the other hand there is even one such property that is not known, then the quantity is unknown.

For instance, if you know that a variable is negative but not its magnitude, can that help solve an equation?

This is very abstract because it depends on what kind of "help" you mean here. It can help in a qualitative manner in general, but you still have to find its value, unless you're given more data about the variable. It may also be the case as mfb points out, that knowing about its sign, will help to choose among different solutions.

 

[Stephen Tashi]

You can solve an equation with one unknown.

That depends on what you mean by "solve". One can "solve" the equation x + 3 = x and obtain the result that there are no solutions.

In speaking of equations, I see no distinction between the term "unknown" and the term "variable". It is actually very difficult to give a precise mathematical definition of a "variable". It would require defining the concept of quantifiers and the concept of the scope of a variable.

This implies a binary distinction between 'known' and 'unknown'.

What implies a binary distinction between "known" and "unknown" ?

But are there grades or shades of unknown?

Are you looking for terminology that's based on the solution(s) of simultaneous equations ?

 

[disregardthat]

The concept of an "unknown", or a variable with "unknown value" is an informal way of speaking. Mathematically, solving an equation, say, $x^2+x-2 = 0$, is simply deducing a statement such as this: "$(\forall x)(x^2 +x - 2 = 0 \Rightarrow (x = 1 \text{ or } x = -2))$". Generally, if $P(x)$ denotes an equation in $x$, then we want to derive a statement on the form $(\forall x)(P(x) \Rightarrow Q(x))$, where $Q(x)$ is some satisfactory formula describing $x$, which we call the solution to $P(x)$. Usually $Q(x)$ is a formula on the form $x = a$, but can also be things like $1 = 0$ which is a false statement in itself. In the latter case we say that $P(x)$ has no solutions. In the former case, we can replace an equation $P(x)$ with a suitable solution formula $Q(x)$.

There is not really a mathematical thing called an "unknown". Formally, it's the usual business with variables, formulas and sentences. We speak of an unknown out of economical reasons, and the fact that writing things out with logical formality is more trouble than it's worth.

 

[pwsnafu]

The concept of an "unknown", or a variable with "unknown value" is an informal way of speaking. Mathematically, solving an equation, say, $x^2+x-2 = 0$, is simply deducing a statement such as this: "$(\forall x)(x^2 +x - 2 = 0 \Rightarrow (x = 1 \text{ or } x = -2))$". Generally, if $P(x)$ denotes an equation in $x$, then we want to derive a statement on the form $(\forall x)(P(x) \Rightarrow Q(x))$, where $Q(x)$ is some satisfactory formula describing $x$, which we call the solution to $P(x)$. Usually $Q(x)$ is a formula on the form $x = a$, but can also be things like $1 = 0$ which is a false statement in itself. In the latter case we say that $P(x)$ has no solutions. In the former case, we can replace an equation $P(x)$ with a suitable solution formula $Q(x)$.

And what about $P(x)$ given as $\sqrt{x^2+x+3} = x$? When you "solve" it you get $Q(x)$ as $x=-3$ which is clearly not the solution.

 

[disregardthat]

And what about $P(x)$ given as $\sqrt{x^2+x+3} = x$? When you "solve" it you get $Q(x)$ as $x=-3$ which is clearly not the solution.

Good observation. Of course, what we do is derive a statement such as $(\forall x)(P(x) \Leftrightarrow Q(x))$ instead. It is usually easy to modify a potential solution $Q(x)$ (for which $P(x) \Rightarrow Q(x)$) in order to get a proper solution $Q(x)$ (for which $P(x) \Leftrightarrow Q(x)$) by testing each potential value of $x$.