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An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra.
The acronym's procedural application does not match experts' intuitive understanding of mathematical notation: mathematical notation indicates groupings in ways other than parentheses or brackets and a mathematical expression is a tree-like hierarchy rather than a linearly "ordered" structure; furthermore, there is no single order by which ...
Brackets are initially around each non-atomic expression, but they can be deleted in cases where there is a defined order of operations, or where order doesn't matter (i.e. where operations are associative). A well-formed expression can be thought as a syntax tree. [41] The leaf nodes are always atomic expressions.
The historical development of mathematical notation can be divided into three stages: [4] [5] Rhetorical stage—where calculations are performed by words and tallies, and no symbols are used. [6] Syncopated stage—where frequently-used operations and quantities are represented by symbolic syntactical abbreviations, such as letters or numerals ...
Quartic equation – an algebraic equation with a degree of four; Quintic equation – an algebraic equation with a degree of five; Polynomial – an algebraic expression consisting of variables and coefficients; Inequalities – a comparison between values; Functions – mapping that associates a single output value with each input value
The use of many symbols is the basis of mathematical notation. They play a similar role as words in natural languages. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.
In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is ...
Use of common words with a derived meaning, generally more specific and more precise. For example, "or" means "one, the other or both", while, in common language, "both" is sometimes included and sometimes not. Also, a "line" is straight and has zero width. Use of common words with a meaning that is completely different from their common meaning.