Faceted Curriculum Project/Syntax

This page defines terms used to describe the language of mathematics in the Faceted Curriculum Project

The Language of Algebra
Using the methods of descriptive linguistics, we identify some lexical and phrasal categories involved in algebraic language.

Lexical Categories

 * 1) The lexical category: quantity, abbreviated $$ Q $$
 * 2) The lexical subcategory: numerical quantity, abbreviated $$ Q_n $$.  Elements of this subcategory include "2", "3.5", "$$\pi$$", and "13.387".
 * 3) The lexical subcategory: variable quantity, abbreviated $$ Q_v $$.  Elements of this subcategory include all letters which are meant to "stand for" numbers, i.e., variables.  In this way, variable quantities have the same relationship to numerical quantities as pronouns do to nouns.


 * 1) The lexical category: relation, abbreviated $$ Rel $$. Elements of this category include "$$=$$", "$$ < $$", and "$$ \geq $$"


 * 1) The lexical category: numerical conjunction, abbreviated $$ Con $$. This includes the usual binary operators such as $$+$$ and $$ \times$$


 * 1) The lexical category: numerical adjective, abbreviated $$ Adj $$. This includes the symbol "-", when used for "negative", as in "-7", as well as the square root symbol "$$\sqrt{}$$".


 * 1) The lexical category: function, abbreviated $$ Fun $$. This includes symbols such as "f", when used to stand for a function, as in "f(x) = x+2".

Phrasal Categories

 * 1) The quantity phrase, abbreviated $$ QP $$. The following are syntactic rules for the formation of quantity phrases:
 * 2) $$ QP \rightarrow Q $$.
 * 3) $$ QP \rightarrow QP Con QP $$.
 * 4) $$ QP \rightarrow Adj Q $$.
 * 5) $$ QP \rightarrow Fun (QP) $$.
 * 6) The algebraic sentence, abbreviated $$ QS $$. The following are syntactic rules for the formation of an algebraic sentence:
 * 7) $$ QS \rightarrow QP Rel QP $$.

Here, we explain the notation used above. A "syntactic rule" such as $$ QP \rightarrow QP Con QP $$ means that a $$ QP $$ (a quantity phrase) may be formed with the sequence "QP, followed by Con, followed by QP". For example, if we know that "3.5" and "2.4" are quantity phrases, and "+" is a conjunction, then "3.5+2.4" is a quantity phrase using the syntactic rule $$ QP \rightarrow QPConQP $$. Note the recursive use of syntactic rules: quantity phrases are often built using other quantity phrases. However, in the end, we not that "3.5" is a quantity phrase, since "3.5" belongs to the lexical category $$Q$$, and we have the rule $$QP \rightarrow Q$$ to make any quantity into a quantity phrase.

Commonly used terms to describe mathematical language

 * 1) An equality is an algebraic sentence, in which the relation is "=".
 * 2) An inequality is an algebraic sentence, in which the relation is "<", ">", "$$\leq$$", or "$$\geq$$".
 * 3) A term is a quantity phrase, in which the every occurring "conjunction" is "$$\times$$" (or a synonym thereof, such as "$$\cdot$$").