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A formal expression is a kind of string of symbols, created by the same production rules as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two formal expressions are considered equal only if they are syntactically equal, that is, if they are the exact same expression.
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.
With some exceptions, an expression can only have components that themselves have components of that type. For example, algebraic expressions can only have constants that are themselves algebraic, and finite sums and products in algebraic expressions can only have algebraic arguments. This template expands to a table.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
The "hierarchy of operations", also called the "order of operations" is a rule that saves needing an excessive number of symbols of grouping.In its simplest form, if a number had a plus sign on one side and a multiplication sign on the other side, the multiplication acts first.
An expression is a finite combination of symbols that is well-formed according to rules that depend on the context. In general, an expression denotes or names a mathematical object, and plays therefore in the language of mathematics the role of a noun phrase in the natural language.