What is the significance of using “$a$” vs “$x$” in this text?

I'm a web development guy currently learning Calculus and am having some trouble understanding the seemingly unwritten rules of variable naming conventions in mathematics.

I've read several other questions relating to "variable naming conventions" here (e.g. laziness on the part of an equation's author, variables are transient, meaningless values, etc.), but I'm still not understanding the motivation driving certain variable name choices.

For instance, I've found certain passages in my various textbooks in which there is an apparent significance in the choice of name an author assigns to a variable. Being unable to understand the thought process involved in examples like the one below is a bit of an impediment to my comprehension of the concepts that are being conveyed.

As a specific example, here is an excerpt from my current textbook, "Calculus: Early Transcendentals," 7th edition, by James Stewart (the guy who built the Calculus house):

In the preceding section, we considered the derivative of a functon f at a fixed number a. $$f'(a) = \lim_{h \to 0} {f(a+h)-f(a)\over h}$$ Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain $$f'(x) = \lim_{h \to 0} {f(x+h)-f(x)\over h}$$

Here, Stewart evidently has some internal distinction between a and x, but it was not mentioned until this point, at which he gives virtually no explanation other than to say that a is a fixed number.

But it looks like a variable, not a fixed number. I get that he is talking about the derivative of a function at a fixed number, like 3, as opposed to a derivative of a function over a range of numbers, but for someone with a software development background, like me, the seemingly arbitrary switch in variable names is confusing.

Is there a somewhat universal convention in mathematics for choosing variable names in different contexts? Or is it just a matter of individual preference, with an explanation of the choice left to the individual author of the equation/formula/expression?

Any help to shed light on this confusing topic would be very much appreciated.

• Letters early in the alphabet often refer to constants and letters late in the alphabet refer to variables. This began with Descartes. – André Nicolas May 22 '15 at 21:33
• @AndréNicolas : But nonetheless, this particular passage that Stewart wrote, the distinction is silly. Either $x$ or $a$ could be any real number at all. The original poster is right about that. In his autobiography Paul Halmos wrote that he once had an instructor who explained that the difference between a function and a value of a function is that a function is something like $f(t)$ and a value of a function is something like $f(\tau)$. He seemed to expect students to say "Oh. Now I understand." ${}\qquad{}$ – Michael Hardy May 23 '15 at 0:49
• @MichaelHardy: I think in this case it is of some pedagogic use. – André Nicolas May 23 '15 at 0:52
• But only if the student is accustomed to the conventions, and very many are not. – Michael Hardy May 23 '15 at 0:53

If you think back to the time when you were in 9th grade learning to solve quadratic equations, what you saw was that $$\text{if } ax^2+bx+c=0\text{ then }x=\frac{-b\pm\sqrt{b^2-4ac\,{}}}{2a}$$ (not to be confused with $\dfrac{-b\pm\sqrt{b^2-4a} c}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-4} ac}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-{}} 4ac}{2a}$, etc., all of which I've seen frequently $\ldots$)

And in individual problems you'd see things like $$2x^2-11x + 9 = 0,\quad\text{so that }a=2,\,b=-11,\,c=9.$$ Thus $a,b,c$ were the things that were known in each particular instance and $x$ was not.

However the way Stewart does this is silly, for reasons pointed out in the original question above.

Here's my favorite example of the fact that understanding the concepts of "constant" and "variable" can matter, and that it depends on context: \begin{align} \frac d {dx} 2^x & = \lim_{h\to0} \frac{2^{x+h}-2^x} h & & \text{(This step you should know by reflex.)} \\[10pt] & = \lim_{h\to0}\left( 2^x \cdot \frac{2^h - 1} h \right) & & \text{(This step is routine algebra.)} \\[10pt] & = 2^x \lim_{h\to 0} \frac{2^h-1} h & & \text{(because $2x$ is }{\bf constant}) \tag 1 \\[10pt] & = \left(2^x \cdot\text{constant}\right) & & (\text{because the limit is }{\bf constant}). \tag 2 \end{align}

Note:

• In line $(1)$, saying $2^x$ is "constant" means it doesn't change as $h$ changes.
• But in line $(2)$, saying the limit is "constant" means it doesn't change as $x$ changes.

"Constant" means not changing as something changes, but what the "something" is can depend on context!

BTW $\dfrac d{dx} e^x = \left(e^x\cdot\text{constant}\right)$, just as above, but the thing that is "natural" about $e$ rather than $2$ as the base is that when the base is $e$ rather than $2$, then the constant is $1$.

• That derivative is an excellent example of the point I was trying to make: it can be difficult for math students to distinguish between letters representing fixed values and those representing variables. I suppose my question should have been, "How can you distinguish between a context in which a letter represents a variable quantity, and a context in which a letter represents a fixed quantity?" It's clearly all about context, but the ability to recognize the context appropriately seems to be a skill that is acquired rather than taught (but that should be taught) in mathematics courses. – tommytwoeyes Jun 7 '15 at 20:29
• I agree: it should be taught. ${}\qquad{}$ – Michael Hardy Jun 8 '15 at 1:30

In mathematics, $x$ usually denotes a variable and $a$ denotes a (fixed) constant (however, any constant).

The idea that the author wanted to give is that if you can calculate the derivative at any point, then you can consider the function that sends each point $x$ to the derivative of $f$ at $x$ (function known precisely as the derivative of $f$).

There's no real difference between $x$ and $a$ used as variable names. We distinguish between variables, parameters, constants - and yet this distinction does not really exist. This hierachy is just a customary order among variables, as it is customary to denote "more variable" variables with $x$ and "more constant" variables with $a$, say. Or to use $n,m,k$ and a few others for integers. Indeed, we could have called the function $x$ and its parameter $f$ without producing anything wrong - it just becomes somewhat illegible due to the readers expectations: $$x'(f)=\lim_{n\to 0}\frac{x(f+n)-x(f)}{n}$$

To give an example that is closer to "your world" maybe: The hostname of a web server need not start with www and that of a mail server need not start with mail. Yet it is customary to run weservices on www.example.com and mail transfer on mail.example.com instead of the other way round. It certainly works fine the other way round but people might get confused.

• Thank you for clarifying that. I thought there must be some naming convention, like the one you described, but none of my textbooks or instructors have covered it. I found some info in the "Variable (mathematics)" article on Wikipedia, but it was less clear than your explanation. I wonder if students who aren't familiar with the concept of naming conventions from another context would know to look for a clarification about the conventions. I really think it's a topic that should be covered in Algebra courses. – tommytwoeyes May 22 '15 at 23:22

Note that, perhaps somewhat counterintuitively, there is no actual difference between "a fixed number $z$" and "a variable $z$". Both mean exactly the same thing: for an arbitrary element $z$ in the domain of $f$, $f'(z)$ is defined to be such-and-such real number. Logically "fixed numbers" and "variables" have the same semantics and correspond to universal quantification (for every $z$ we do the following / the following holds).

The difference is in pragmatics, that is, in what purpose we want to signal to the reader (you can perhaps think of this as defining multiple type synonyms for the same underlying type - you make up creative names for basic types to, among other things, better describe what the code does).

"A fixed number $a$" signals that the reader is supposed to concentrate on values $f'(a)$ one at the time, and the choice of name reinforces this wording. "A variable $x$" means that the reader should think about $f'(x)$ changing as $x$ does, and about the function $f'$.

As for the appropriate choice of names, it's impossible to give hard-and-fast rules, but you get used to various conventions subconsciously, and others have already given some useful guidelines.

edit: I overlooked that Hagen von Eitzen's answer makes the same point about variables and constants being technically the same thing, so this is mostly a duplicate of his answer.

Usually the first letters $a,b$ are used to indicate constant values, that are not specified but are intended to be fixed.

The last letters $x,y,z$ are used to indicate variables, that is a symbol for a number that can have any value. In an equation we usually want to find the value of the variables, considering the other terms as fixed.

But this rule is not so rigid, and we have to be careful to the contest to understand what are constant or variables.

In many cases the constants can vary in a range (are called parameters) and in this case ve can have problems where we want to find the values of the parameters in such a way that the variables have some given value.

In your case you can think at $a$ in the first equation as a fixed number and $h$ as a variable in a rang that is a neigborough of $0$. In the second equation $x$ is a variable and you can think $h$ as a parameter.

• Thank you very much. This is just the sort of clarification I was hoping to find. – tommytwoeyes May 22 '15 at 23:13
• @Emilio_Novati How can you determine when a letter, such as x or h is used as a variable, parameter, or a fixed number, as they are used in different contexts in the two equations, like you mentioned? This is precisely the sort of distinction I was trying to articulate, which seems important to the understanding the limit concept but is a difficult distinction to make unless someone points it out, as you have. – tommytwoeyes May 22 '15 at 23:18
• A parameter is a kind of variable that we have to determine so that some conditions are verified. So the difference is not so well defined and dependent contest and the problem we want to solve. Also in a simple expression as $ax+b=0$ the terms $a \ne 0$ and $b$ can be constant if we want simply solve the equation in $x$, but if we search for what values of $a$ the equation has solutions $x>0$ than $a$ and $b$ are parameters. – Emilio Novati May 23 '15 at 18:26

Say you have a program p that returns the input plus one.
it could be defined like this:

p(int i){
i=i+1;
return i;
}

p(int i) is like f'(x): it's a function that takes one input. Now, say you have another program that calls p at some point. for instance,

q(int i, int j){
s=i+j;
return p(s);
}

p(s) is like f'(a). It's a specific number that depends on the value of s.
The difference is subtle but it's there; by convention, x and the like are often used to indicate an "empty" variable, while a and the like are used to express "filled" variables. In both cases you have to write a letter, just as you have to do so in a program.

In here, the author talks about the filled variable first; it's a bit like when at first you do a calculation inside a program, writing it out each time, and then you decide to write a function outside the program and call that when you want to do those steps.

• I see - so you're saying the author has created an abstraction of the first function in the second function. Ok, that makes more sense. Thanks. It is a subtle difference, but when you put it in that context, it helps. – tommytwoeyes May 23 '15 at 0:19
• I'd like to say "you got it! Glad it helped!", but I got a downvote, so maybe I was misleading? I hope not. – byserpas May 23 '15 at 8:02