expressions

An expression is a mathematical phrase that combines numbers, letters, and operations like adding or multiplying, but it does not say two things are equal — so it has no equals sign.

What an expression really is

Think of an expression as a recipe rather than a finished dish. Something like $3x + 5$ tells you what to do — "take a number, multiply it by 3, then add 5" — without telling you the final answer yet. The letter $x$ is a variable, a placeholder standing in for a number we don't know or that can change.

Compare this to an equation, which does have an equals sign, like $3x + 5 = 20$. An equation is a claim; an expression is just a description. This is a bit like the difference between a shopping list ("apples plus bread") and a receipt saying the total is $20.

The parts of an expression

Expressions are built from a few simple pieces:

• Terms — the chunks separated by $+$ or $-$ signs. In $3x + 5$, the terms are $3x$ and $5$.
• Coefficients — the number multiplying a variable. In $3x$, the coefficient is $3$.
• Constants — plain numbers that never change, like the $5$.

Leonhard Euler's Elements of Algebra (1770), an early textbook that taught working with symbolic expressions — source

Where the idea came from

The word algebra comes from the Arabic al-jabr, meaning "reunion of broken parts." It appears in the title of a book written around 820 CE by the Persian scholar Muhammad ibn Musa al-Khwarizmi in Baghdad. His name, latinized, also gives us the word algorithm. Back then, expressions were written out in full sentences. The neat symbolic style we use today — with letters for unknowns — was popularized much later by the French mathematician François Viète in the 1590s.

Why it matters

Expressions are the vocabulary of algebra. Before you can solve problems, you must first describe them. A phone plan costing $10 plus $2 per gigabyte becomes $10 + 2g$ — a tidy expression that captures every possible bill in one line.

Further exploration

• Leonhard Euler, Elements of Algebra (1770) — a famously clear introduction, written to teach a beginner from scratch.
• Al-Khwarizmi, The Compendious Book on Calculation by Completion and Balancing (c. 820) — the founding text that named the whole subject.