Expression (mathematics)

In mathematics, an expression is a way to write numbers, operations, and symbols together following special rules. These symbols can stand for numbers, unknown values called variables, actions like adding or multiplying, and even functions, which are rules for changing numbers. Punctuation and brackets help organize these pieces so we know the order to do things.

Expressions are different from formulas. While expressions represent numbers or values, formulas make statements about those values. For example, “8 times x minus 5” is an expression, but “8 times x minus 5 is greater than or equal to 3” is a formula because it tells us something about the value.

We can work with expressions by simplifying them or finding their value. For instance, the expression “8 times 2 minus 5” simplifies to “16 minus 5,” which equals 11. Expressions are also used to describe functions, which are rules that take an input number and give us an output number using the expression. For example, the expression “x squared plus 1” defines a function that tells us what we get when we square a number and then add one.

Elementary mathematics

In elementary algebra, a variable is a letter that stands for a number that can change. When we evaluate an expression, we replace the variable with a specific number and calculate the result. We can also simplify expressions by combining similar parts or performing calculations step by step.

For example, in the expression 4 x 2 + 8, if we let x = 3, we first calculate 3² (which is 9), then multiply by 4 to get 36, and finally add 8 to get 44. Expressions can include numbers, variables, and operations like addition or multiplication. When expressions have the same variables raised to the same powers, we can combine them to make the expression simpler.

Well-defined expressions

Main article: Well-defined expression

In math, expressions are ways to write numbers, operations, and ideas using symbols. For an expression to be correct, it needs to follow certain rules. These rules are about how symbols can be put together, which we call syntax. For example, "1 + 2 × 3" is correct, but a messy string of symbols like "× 4) x +, / y" is not.

Besides being correctly written, an expression also needs to have a clear meaning. This is called semantics. An expression with a clear meaning is well-defined. For example, "1 ÷ 0" looks right but does not have a meaning, so it is not well-defined. Well-defined expressions give one clear result, like how "a × b × c" always means the same thing no matter how you multiply the numbers together.

Formal definition

The term 'expression' is part of the language of mathematics. It is not defined within mathematics itself but is a basic part of the language we use to talk about math.

In math, an expression is made up of symbols like numbers, variables (letters that stand for unknown numbers), operations (like addition or multiplication), and brackets (which help us know the order to do things in). Simple expressions can be just a single number or variable, like "2" or "x". More complex expressions combine these using operations, like "3 + 4" or "x × y". Brackets help organize these combinations, such as in "(2 + 3) × 4".

Computer science

Main article: Expression (computer science)

In computer science, an expression is a combination of constants, variables, functions, and operators. The programming language interprets these parts and computes them to produce a new value. This process is called evaluation. The result can be a number, a string, or a Boolean value, among other types.

Expressions are different from statements, which are instructions that do not return a value. In computer algebra, expressions can be evaluated based on the values given to their variables. For example, the expression "8 x − 5 ≥ 3" is false if x is less than 1, and true otherwise. Expressions can also represent equations and matrices in computer algebra software.

Types of expressions

An algebraic expression is built from numbers, variables, and operations like addition, subtraction, multiplication, division, and exponentiation. For example, (3x^2 - 2xy + c) is an algebraic expression. Polynomials are special types of algebraic expressions made from numbers, variables, and operations of addition, multiplication, and exponentiation with whole number exponents. For instance, (3(x + 1)^2 - xy) is a polynomial.

Formal expressions are sequences of symbols created by specific rules, used without considering their meaning. Two formal expressions are equal only if they look exactly the same. For example, the formal expressions “2” and “1+1” are not considered equal.

History

For broader coverage of this topic, see History of mathematics and History of mathematical notation.

See also: History of the function concept

Early written mathematics

The earliest written math began with simple tally marks carved into wood or stone. One famous example is the Ishango bone, found near the Nile and dating back over 20,000 years ago. It shows early counting methods. Ancient Egypt used symbols for numbers and basic math operations, recorded in texts like the Rhind Mathematical Papyrus. In Mesopotamia, numbers were written in a base-60 format on clay tablets, a system we still use today for time and angles.

Syncopated stage

The "syncopated" stage introduced shortcuts for common math operations. Ancient Greek mathematics focused mostly on geometry but Diophantus of Alexandria began using symbols to represent unknown numbers and their powers. He used special symbols for operations like squaring and cubing numbers.

Symbolic stage and early arithmetic

The move to fully symbolic algebra began with mathematicians like Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī. Symbols like the plus sign (+) and minus sign (−) were introduced. René Descartes helped formalize the use of letters for variables, and later Isaac Newton and Gottfried Wilhelm Leibniz developed calculus.