Algebraic expression

"Rational expression" redirects here. For the notion in formal languages, see regular expression.

In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).^[1] For example, $3x^2 - 2xy + c$ is an algebraic expression. Since taking the square root is the same as raising to the power $\tfrac{1}{2}$ ,

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

is also an algebraic expression. By contrast, transcendental numbers like π and e are not algebraic.

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, $\frac{3x^2 - 2xy + c}{y^3-1}$ is a rational expression, whereas $\sqrt{\frac{1-x^2}{1+x^2}}$ is not.

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.

Terminology

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

1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, $x, y$ - variables

In roots of polynomials

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 $\ge$ 5.

Conventions

Variables

By convention, letters at the beginning of the alphabet (e.g. $a, b, c$ ) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x, y$ and $z$ ) are used to represent variables.^[2] They are usually written in italics.^[3]

Exponents

By convention, terms with the highest power (exponent), are written on the left, for example, $x^2$ is written to the left of $x$ . When a coefficient is one, it is usually omitted (e.g. $1x^2$ is written $x^2$ ).^[4] Likewise when the exponent (power) is one, (e.g. $3x^1$ is written $3x$ ),^[5] and, when the exponent is zero, the result is always 1 (e.g. $3x^0$ is written $3$ , since $x^0$ is always $1$ ).^[6]

Algebraic vs. other mathematical expressions

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions.

	Arithmetic expressions	Polynomial expressions	Algebraic expressions	Closed-form expressions	Analytic expressions	Mathematical expressions
Constant	Yes	Yes	Yes	Yes	Yes	Yes
Variable	Yes	Yes	Yes	Yes	Yes	Yes
Elementary arithmetic operation	Yes	Yes	Yes	Yes	Yes	Yes
Factorial	Yes	Yes	Yes	Yes	Yes	Yes
Integer exponent	No	Yes	Yes	Yes	Yes	Yes
N-th root	No	No	Yes	Yes	Yes	Yes
Rational exponent	No	No	Yes	Yes	Yes	Yes
Irrational exponent	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
Gamma function	No	No	No	No	Yes	Yes
Bessel function	No	No	No	No	Yes	Yes
Special function	No	No	No	No	Yes	Yes
Continued fraction	No	No	No	No	Yes	Yes
Infinite series	No	No	No	No	Yes	Yes
Formal power series	No	No	No	No	No	Yes
Differential	No	No	No	No	No	Yes
Limit	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 $\sqrt x + 4$ .

Notes

↑ Morris, Christopher G. (1992). Academic Press dictionary of science and technology. p. 74.
↑ William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71
↑ James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, ISBN 0387985425, 9780387985428, 221 pages, [James E. Gentle page 183]
↑ David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, ISBN 0470185597, 9780470185599, 304 pages, page 72
↑ John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 31
↑ Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, ISBN 0538733543, 9780538733540, 803 pages, page 222

References

James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. p. 8.

External links

Weisstein, Eric W., "Algebraic Expression", MathWorld.

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