The Letter That Holds a Number

A variable is a mystery number wearing a name tag. Feed the machine any number n, and it always does the same thing — say, double it and add one: 2n + 1.

Try it: if n is 4, the machine gives 2 × 4 + 1 = 9. If n is 10, it gives 21. One rule, endless numbers — that is the whole trick.

A variable is a letter that stands for a number. It is not a mystery forever — it is a placeholder, like the blank in __ + 3 = 10, except we can do algebra with it. An expression combines variables and numbers with operations: 3n + 2 means three times n, then add two.

Two conventions to read fluently: writing a number against a letter means multiply (3n is 3 × n), and a letter by itself has a silent 1 in front (n means 1n). To evaluate an expression, substitute a value and compute: at n = 5, 3n + 2 = 3(5) + 2 = 17.

A variable names a quantity that may vary; an expression is a recipe built from variables, constants, and operations. The value of 3n + 2 depends entirely on n — substitute n = 5 and the expression evaluates to 17. The parts have names worth owning: in 3n + 2, the 3 is a coefficient, the 2 a constant term.

One letter can appear many times, and each appearance means the same number: in n² + 3n, both n's move together. Different letters may hold different values — or happen to hold the same one. This discipline, one name per quantity, is what lets algebra scale from tickets to orbital mechanics.

Here is a box. Some marbles are hiding inside. We do not know how many. Let's call the hidden number n.

If two more marbles drop in, the box holds n + 2. We can say that even before we peek.

Formally, a variable is a name ranging over a set: when we write 3n + 2 with n ∈ ℝ, we define a function ℝ → ℝ rather than a single number. Evaluation is the act of applying that function at a point. Much of early algebra is learning to treat the expression itself — not its value — as the object of study.

The distinction matters: 3n + 2 and 3(n + 1) − 1 are different expressions but the same function — they agree at every input. Algebraic manipulation is the art of moving between equivalent expressions while preserving the function underneath.

In the syntax of a formal language, variables are symbols; terms like 3n + 2 are finite trees over a signature (here: constants 3, 2 and binary operations ·, +). Semantics enters through a valuation ν assigning elements of a structure to variables; the term then denotes via recursive interpretation. School algebra's 'substitute and compute' is exactly ⟦t⟧ν.

The free/bound distinction — invisible in Algebra I but latent — arrives the moment a quantifier or Σ binds a letter. That n in 3n + 2 is free is precisely why the expression defines a function of n. Keeping syntax and semantics separate is the mature form of this lesson's box-with-a-number picture.