This chapter showed that a letter can do at least five different jobs and that the writing hardly ever says which. The guide below gathers the questions that help to find out.

Five questions for a letter

  1. Does it hide a value or stand for any value? If it hides one, it is an unknown, and the task is to find it. If it stands for any, the sentence is a general law or an identity.
  2. Does it move during the problem? If so, it is a variable, and it is worth asking what it depends on and what depends on it.
  3. Does it stay still while the others move? Then it is a parameter or a constant. It is worth asking: fixed for ever, like $\pi$, or only in this problem, like the $a$ in $ax^2$?
  4. Is it tied to an operator? If it appears in a $\Sigma$, an $\int$, a $\forall$ or a $\mapsto$, it is a bound letter, and it can be swapped for another without changing anything.
  5. Which word in the text announces the role? “Find”, “for every”, “let”, “when”, “given”. If none appears, the conventions of the subject decide, and it is good to make them explicit.

A full reading of y = ax² + bx + c

  1. $y$ and $x$: variables. $x$ is the independent one, $y$ the dependent one. Together they run through the points of a parabola.
  2. $a$, $b$ and $c$: parameters. Fixed during the problem, they choose which parabola. The $a$ must be non-zero, or the curve becomes a straight line.
  3. Change the question and the roles change. In “where does the parabola cross the $x$-axis?”, we set $y=0$, and $x$ becomes an unknown. In “which parabola passes through $(1,2)$, $(2,3)$ and $(3,6)$?”, the parameters become the unknowns, and $x$ and $y$ become data: $a+b+c=2$, $4a+2b+c=3$, $9a+3b+c=6$, which give $a=1$, $b=-2$, $c=3$.
  4. No letter is a dummy. They all matter: swapping $a$ for $k$ requires swapping it throughout the text.

One short line, three different roles, and each question rearranges everything.

Glossary

  • Unknown. A letter that stands for a definite value, not yet known, fixed by the conditions of the problem. In $2x+1=7$, $x=3$.
  • Generalised number. A letter that stands for any element of a set, in a sentence that holds for all of them. In $a+b=b+a$.
  • Identity. An equality true for all values of the letters, such as $(a+b)^2=a^2+2ab+b^2$. It is not solved; it is proved.
  • Variable. A letter that runs through the values of a set, usually together with other letters on which it depends or which depend on it.
  • Independent and dependent variable. In $y=f(x)$, $x$ is chosen freely and $y$ is determined by the rule. The choice may be reversed depending on the question.
  • Parameter. A letter fixed during a problem, but which may take different values from one problem to another. It picks out one member of a family.
  • Constant. A letter whose value is always the same, such as $\pi$, $e$ or the speed of light $c$.
  • Free letter. A letter on which the value of an expression depends. In $\sum_{k=1}^n k$, the $n$.
  • Bound (or dummy) letter. A letter tied to an operator, which can be swapped for another without changing the meaning. In $\sum_{k=1}^n k$, the $k$.
  • Function. A rule that assigns to each element of one set, the domain, a single element of another. $f$ is the function; $f(x)$ is its value at $x$.
  • Variable capture. The mistake of using the same letter as free and as bound in the same expression, as in $\int_0^x x\,dx$.
Exercise: apply the guide

Read $\displaystyle V(r)=\frac43\pi r^3$. $V$ is the name of a function, and $V(r)$ its value. $r$ is the independent variable, the radius. $\pi$ is a constant. $\frac43$ is a number. There are no bound letters. If the question is “what radius gives a volume of $36\pi$?”, $r$ becomes an unknown: $\frac43r^3=36$, $r^3=27$, $r=3$.

Exercise: find the hidden bound letter

In the definition $f'(x)=\displaystyle\lim_{h\to0}\frac{f(x+h)-f(x)}h$, which letter is bound? The $h$: the limit runs through it and makes it disappear, and swapping it for $\Delta x$ or for $t$ changes nothing. The $x$ is free, and the derivative depends on it. The $f$ is the name of a given function.

Chapter index · ← When the letter misleads: students, professors and apples · References and credits →