Expression (mathematics)

An expression in mathematics is a way to write numbers, operations, and symbols together. These symbols can stand for numbers, unknown values called variables, actions like adding or multiplying, and functions. Punctuation and brackets help organize these pieces.

Expressions are different from formulas. Expressions represent numbers or values, while 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.

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 an output number using the expression. For example, the expression “x squared plus 1” defines a function.

Elementary mathematics

In elementary algebra, a variable is a letter that stands for a number. When we evaluate an expression, we replace the variable with a number and find the result. We can also simplify expressions by combining parts or doing 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 and operations using symbols. For an expression to be correct, it must follow certain rules. These rules tell us how symbols can be put together. For example, "1 + 2 × 3" is correct.

An expression also needs to have a clear meaning. This is called being well-defined. For example, "1 ÷ 0" looks right but does not have a clear meaning, so it is not well-defined. Well-defined expressions give one clear result.

Formal definition

The term 'expression' is part of the language of mathematics. It is a basic part of how we talk about math.

In math, an expression is made up of symbols. These can be numbers, variables (letters that stand for unknown numbers), operations (like addition or multiplication), and brackets (which tell us the order to do things). 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 mix of constants, variables, functions, and operators. The programming language looks at these parts and works them out to give a new value. This is called evaluation. The result can be a number, a string, or a Boolean value, among other types.

Expressions are not the same as statements. Statements are instructions that do not give back a value. In computer algebra, expressions can be worked out using 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 show equations and matrices in computer algebra software.

Types of expressions

An algebraic expression is made from numbers, letters that stand for numbers (called variables), and math actions like addition, subtraction, multiplication, division, and exponentiation. For example, (3x^2 - 2xy + c) is an algebraic expression.

Polynomials are a special kind of algebraic expression. They use numbers, variables, and only addition, multiplication, and exponentiation with whole number exponents. For example, (3(x + 1)^2 - xy) is a polynomial.

Formal expressions are sets of symbols put together by exact rules. They are used without thinking about what they mean. Two formal expressions are only the same if they look exactly alike. 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 started 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, 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 brought shortcuts for common math operations. Ancient Greek mathematics focused mostly on geometry, but Diophantus of Alexandria began using symbols for unknown numbers and their powers. He used special symbols for operations like squaring and cubing numbers.

Symbolic stage and early arithmetic

The shift to 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 using letters for variables, and later Isaac Newton and Gottfried Wilhelm Leibniz developed calculus.