In arithmetic you work with particular numbers: 2 + 3 = 5, 4 + 1 = 5. Algebra is the next step — instead of one fixed number, we use a letter to stand for any number. A letter like a or n is a placeholder: it holds a spot that any number could fill.

That one idea is surprisingly powerful. It lets us write a single statement that captures a whole pattern at once, instead of checking number after number forever.

We already met a fact about the laws of arithmetic: swapping the order of an addition never changes the answer. With numbers we can only ever show examples — 2 + 3 = 3 + 2, 4 + 1 = 1 + 4, and so on. But there are infinitely many pairs, so we can never finish the list.

Algebra lets us say it once, for every pair at the same time. Press play to watch two number facts fade into letters and become a single general rule:

Once a pattern is written with letters, we can do things with it. Algebra is the toolkit for working with these general statements: we can apply the order of operations to expressions full of letters, simplify them, solve equations to find an unknown number, and even draw expressions as graphs. The same letters happily stand for fractions or negative numbers too — a rule like a + b = b + a holds for every kind of number, not just the counting numbers.

See it explained

Khan Academy explains why algebra is full of letters in the first place: