In mathematics, an **algebraic expression** is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).^{[1]} For example, 3*x*^{2} − 2*xy* + *c* is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression:

An *algebraic equation* is an equation involving only algebraic expressions.

By contrast, transcendental numbers like π and e are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, π is constructed as a geometric relationship, and the definition of e requires an *infinite number* of algebraic operations.

A **rational expression** is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic. Thus,

is a rational expression, whereas

is not, i.e. this is an irrational expression.

A **rational equation** is an equation in which two rational fractions (or rational expressions) of the form

are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.

Algebra has its own terminology to describe parts of an expression:

The roots of a polynomial expression of degree *n*, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if *n* < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if *n* 5.

By convention, letters at the beginning of the alphabet (e.g. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and ) are used to represent variables.^{[2]} They are usually written in italics.^{[3]}

By convention, terms with the highest power (exponent), are written on the left, for example, is written to the left of . When a coefficient is one, it is usually omitted (e.g. is written ).^{[4]} Likewise when the exponent (power) is one, (e.g. is written ),^{[5]} and, when the exponent is zero, the result is always 1 (e.g. is written , since is always ).^{[6]}

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions.

Arithmetic expressions | Polynomial expressions | Algebraic expressions | Closed-form expressions | Analytic expressions | Mathematical expressions | |
---|---|---|---|---|---|---|

Constant | Yes | Yes | Yes | Yes | Yes | Yes |

Elementary arithmetic operation | Yes | Addition, subtraction, and multiplication only | Yes | Yes | Yes | Yes |

Finite sum | Yes | Yes | Yes | Yes | Yes | Yes |

Finite product | Yes | Yes | Yes | Yes | Yes | Yes |

Finite continued fraction | Yes | No | Yes | Yes | Yes | Yes |

Variable | No | Yes | Yes | Yes | Yes | Yes |

Integer exponent | No | Yes | Yes | Yes | Yes | Yes |

Integer nth root | No | No | Yes | Yes | Yes | Yes |

Rational exponent | No | No | Yes | Yes | Yes | Yes |

Integer factorial | No | No | Yes | Yes | Yes | Yes |

Irrational exponent | No | No | No | Yes | Yes | Yes |

Exponential function | No | No | No | Yes | Yes | Yes |

Logarithm | No | No | No | Yes | Yes | Yes |

Trigonometric function | No | No | No | Yes | Yes | Yes |

Inverse trigonometric function | No | No | No | Yes | Yes | Yes |

Hyperbolic function | No | No | No | Yes | Yes | Yes |

Inverse hyperbolic function | No | No | No | Yes | Yes | Yes |

Root of a polynomial that is not an algebraic solution | No | No | No | No | Yes | Yes |

Gamma function and factorial of a non-integer | No | No | No | No | Yes | Yes |

Bessel function | No | No | No | No | Yes | Yes |

Special function | No | No | No | No | Yes | Yes |

Infinite sum (series) (including power series) | No | No | No | No | Convergent only | Yes |

Infinite product | No | No | No | No | Convergent only | Yes |

Infinite continued fraction | No | No | No | No | Convergent only | Yes |

Limit | No | No | No | No | No | Yes |

Derivative | No | No | No | No | No | Yes |

Integral | No | No | No | No | No | Yes |

A *rational algebraic expression* (or *rational expression*) is an algebraic expression that can be written as a quotient of polynomials, such as *x*^{2} + 4*x* + 4. An *irrational algebraic expression* is one that is not rational, such as √*x* + 4.