These two expressions have the same value:

$$\sum_{k=1}^{4}k^2=1+4+9+16=30,\qquad\sum_{j=1}^{4}j^2=1+4+9+16=30.$$

Swapping $k$ for $j$ changed nothing. The letter is there only to mark where the calculation passes; once the sum is done, it disappears. Letters like these are called dummy, or bound. They are the opposite of unknowns: an unknown keeps a value that matters; a dummy letter keeps no value at all once the operation is over.

Free and bound

Now compare:

$$\sum_{k=1}^{n}k=\frac{n(n+1)}{2}.$$

Here there are two letters of different kinds. The $k$ is bound by the sum: it exists only between the $\Sigma$ and the end of the expression, and could be swapped for any other letter. The $n$ is free: the result depends on it. Swapping $n$ for $m$ gives another statement, about another number.

The same happens in the integral:

$$\int_0^x t\,dt=\frac{x^2}{2}.$$

The $t$ is bound by the integral, and the $dt$ marks its scope, like a closing bracket. The $x$ is free: the area depends on how far we integrate. And in the quantifiers of logic:

$$\forall x\,(x+0=x)$$

the $x$ is bound by the $\forall$, and the whole sentence depends on no value at all: it is simply true.

Operator Bound letter Scope
$\displaystyle\sum_{k=1}^{n}$ $k$ the term that follows
$\displaystyle\int_a^b\cdots\,dt$ $t$ between the $\int$ and the $dt$
$\forall x,\ \exists x$ $x$ the formula that follows
$\lim\limits_{h\to0}$ $h$ the expression under the limit
$\{x\mid x>2\}$ $x$ inside the braces
$x\mapsto x^2$ $x$ the rule of the function

The mistake of capturing the letter

Because a dummy letter can be swapped, there is a dangerous temptation: to use the same letter for two things. The formula

$$\int_0^x x\,dx$$

is a trap. The $x$ in the upper limit is free; the $x$ inside the integral is bound. The expression seems to say “integrate $x$ from $0$ to $x$”, but it is not clear which $x$ is which. The correct form separates the letters: $\int_0^x t\,dt=\frac{x^2}2$. In logic and in programming this mistake has a name, variable capture, and compilers of programming languages rename variables automatically to avoid it.

Exercise: free or bound?

In $\displaystyle\sum_{k=1}^{m}(k+a)$, which letters are free? The $k$ is bound by the sum. The $m$ and the $a$ are free. Indeed, the result depends only on them: $\sum_{k=1}^{m}(k+a)=\frac{m(m+1)}2+ma$. Swapping $k$ for $j$ does not change the value; swapping $a$ for $b$ does.

Exercise: fix the integral

A student wrote $\displaystyle F(x)=\int_0^x 3x^2\,dx$. Rewrite it with separate letters and compute it. The correct form is $F(x)=\int_0^x3t^2\,dt=x^3$. With the student’s writing, someone could treat $3x^2$ as a constant, because the “outside” $x$ is fixed, and get $3x^2\cdot x=3x^3$, which is wrong.

Chapter index · ← f, f(x) and x ↦ x²: three ways to name a function · When the letter misleads: students, professors and apples →