algebraic notation

Algebraic notation is the shorthand that lets you write a mathematical idea in a few symbols instead of a paragraph of words. It is to mathematics what musical notation is to music: a compact, agreed-upon way to record exactly what is meant, so that anyone who knows the conventions reads the same thing.

Its conventions are economical and worth knowing explicitly. Multiplication is shown by writing letters next to each other (ab means a times b) and the number always comes first (3x, not x3). A power is written with a small raised number (x^2). A variable with no visible coefficient has a coefficient of 1, and grouping symbols — parentheses, brackets — control what is done first. These habits remove ambiguity.

This notation grew slowly. Symbols we take for granted — the equals sign (Robert Recorde, 1557), the + and − signs, the use of x for the unknown — were each invented at some point and then adopted because they made reasoning easier. Good notation is not decoration; by hiding routine detail it frees the mind to see structure, which is why a well-chosen symbol can be worth a thousand words.