Question about variable and constant notation in some properties

Question

I am just starting to think about mathematical notation a lot more and some parts of it do not make as much sense to me as I would like. I am operating under the convention that beginning alphabet letters are constants and ending alphabet letters are variables.

I see this equation in algebra and calculus text books: $a(b+c)=ab+ac$ What this means to me is that it is true for constants, but says nothing about variables. But this clearly is true for variables as well, as you can do equations such as $x^2-2x=x(x-2)$.

Why is some math notation written this way, or rather, why isn't the property written as $x(y+z)=xy+xz$ since that would signify that it works for any unknown and not just constants. This goes for all the elementary algebra fraction equations as well. And then to make it even more confusing, as soon as exponents are involved, variables are now introduced, such as $x^mx^n=x^{m+n}$. Wouldn't a constant work just as well here? Am I just not understanding something basic about notation?

p.s. If there is a good resource to clarify mathematical notation I would love to see it.

Answer 1

Of course, they are all variables. Your perception is somewhat correct in the sense that customarily constants (or, "parameters") in the context of a specific problem are often denoted $\{a,b,c,\ldots\}$. But a symbol is a symbol is a symbol. Even the symbol $\pi$ isn't always used just to mean the number $3.1415926535...$

Answer 2

I don't quite get you. When you say $a(b+c) = ab +bc$ is true for 'constants', does that mean it is true not only for $a=1$, $b=2$ and $c=3$, but also for $a=5$, $b=3$ and $c=3$, and so on and so forth. So there you are, it is varying in that sense. When I have an equation, say $y=x²+c$ then only $x$ and $y$ are varying and $c$ is fixed, but $c$ can be any 'constant'. It's not varying in the sense in which $x$ and $y$ are varying. So depends on what the context is, I suppose.

Answer 3

Actually, in a formula such as $ax^2 + bx + c = 0$, we might say $a$, $b$, and $c$ are parameters and treat them as constants, but when we write the formula to solve for $x$, we are implicitly saying this formula works for any values of $a$, $b$, and $c$ (or if we are working in real numbers only, any values that satisfy a certain condition), not just for three constants that we happened to have in mind. If we wanted to write a more formal statement of how to find $x$, we might start to use quantifiers such as $\forall$, and then $a$, $b$, and $c$ would formally be called variables (but variables bound to the quantifiers, which is still different from the free variable $x$). In contrast, the $2$ and $4$ that appear in the quadratic formula really are constants; we can't change one of them to $5$ and expect that the formula will still be valid.

And yes, I remember this as a huge point of confusion for second-year calculus students when some parameters suddenly started acting like variables in the middle of a lecture. Perhaps you can say the symbols are all variables, but some are more variable than others.