Algebraic expression

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:

{\sqrt {\frac {1-x^{2}}{1+x^{2}}}}

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,

{\frac {3x-2xy+c}{y-1}}

is a rational expression, whereas

{\sqrt {\frac {1-x^{2}}{1+x^{2}}}}

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

{\frac {P(x)}{Q(x)}}

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.

^ Morris, Christopher G. (1992). Academic Press dictionary of science and technology. Gulf Professional Publishing. p. 74. algebraic expression over a field.

[1] Morris, Christopher G. (1992). Academic Press dictionary of science and technology. Gulf Professional Publishing. p. 74. algebraic expression over a field.

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