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Posted 7 July 2011

The Symbolic Language of Math

The symbolic language of math is a distinct special-purpose language.  Unlike mathematical English, it is not a variety of English.  It has its own rules of grammar that are quite different from those of English.  You can usually read expressions in the symbolic language in any math article written in any language. 

The chapter More about the languages of math discusses topics that involve both the symbolic language and mathematical English.

Symbolic expressions

The symbolic language consists of symbolic expressions written in the way mathematicians traditionally write them.   A symbolic expression consists of symbols arranged according to specific rules.

Symbolic expressions occur in two types:

¨  Symbolic assertions.  These are complete statements that stand alone as sentences.   An assertion says something.  An assertion may contain variables and be true for some values of the variables and false for others.  Assertions play the same role in the symbolic language as assertions in math English.  Following the usage under math English, a symbolic statement is a symbolic assertion that is either true or false.

¨  Symbolic terms. They are expressions that refer to some mathematical object.   A term names something.  Terms play the same role in the symbolic language that names do in math English.  See remark for variations in usage of the word “term”.


Every expression in the symbolic language is either a symbolic term or a symbolic assertion.           




¨  "\(\pi >0\)". This is a symbolic statement  in the symbolic language of mathematics. It is true.  “$\pi <0$” is a false symbolic statement.

¨  The expression


¨   is a symbolic assertion.   It is true for some values of x and y and false for others.  For example, it is false for x = 1 and y = 0 and true for x = 1 and y = 2.  See more about symbolic assertions like this one in the section on constraints


¨   is also a symbolic assertion.  It is true for x = 1 and x = 4 and false for all other values of x.

¨  The expression “\({{3}^{2}}\)” is a symbolic term.  It is another name for the number 9. 

¨  The expression “<” is a symbolic term.  It refers to the less-than relation, which is a mathematical object.

¨  “${{\sin }^{2}}x$” (which means $\sin x\cdot \sin x$)  is a symbolic term containing a variable x, so it has variable meaning depending on which value is substituted for x.  For example, ${{\sin }^{2}}\left( \pi /4 \right)$ is another name for 1/2.