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Produced by Charles Wells     Revised 2017-01-22

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Symbolic expressions

Reading symbolic expressions

Images and metaphors for symbolic expressions

Grammar of the symbolic language

Variables and substitution

Variable objects



Other symbols

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.

This chapter discusses aspects of the symbolic language that may cause difficulties to newcomers. It is not a systematic introduction to the symbolic language. You can find more information in the links in The languages of math.

Warning: The terminology used here to talk about symbolic expressions is nonstandard. See Variations in terminology for more detail.

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. Every symbolic expression is one of two types: symbolic assertion and symbolic term.

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

Symbolic assertions

A symbolic assertion is a complete statement that stands alone as a sentence.

A symbolic assertion says something.


Symbolic statements

A symbolic statement is a symbolic assertion without variables.


Symbolic terms

A symbolic term is a symbolic expression that refers to some mathematical object.

A symbolic term names something.

Terms play the same role in the symbolic language that descriptions do in math English. 


Variations in terminology

The names “symbolic assertion” and “symbolic term” are not standard usage in math.  In mathematical logic:

All these words, as well as my use of “term”, can cause cognitive dissonance:

This kind of conflict between different parts of math
happens all the time.
Neither side is right or wrong. Get used to it. 

Non-algebraic expressions

Symbolic expressions don’t have to have algebraic form and they do not have to name numbers.


Each branch of mathematics is concerned with certain particular kinds of mathematical objects, and every one of them studies many different kinds of operations on the objects, expressed (usually) in symbolic notation.

Reading symbolic expressions

Distinguish between assertions and terms

A fundamental difficulty many people new to algebra have is that they don't pay attention to the difference betweeen assertions and terms.


An expression such as “$x\alpha y$, where $\alpha$ is any old symbol, may be an assertion (saying something) or a term (naming something).

Division and fractions

Two symbols used in the study of integers are notorious for their confusing similarity.

Notice that $m/n$ is an integer if and only if $n|m$. Not only is $m/n$ a number and $n|m$ a statement, but the statement "one is an integer if and only if the other is true" is correct only after the $m$ and $n$ are switched!

It is wise to be a bit paranoid
about whether you really understand
a particular kind of math notation.

Be patient

When you see a complicated assertion or term you have to be patient. You must stop and unwind it.  Read the tiresomely long example of unwinding an expression in Zooming and Chunking.

Giving names to symbolic expressions

Turning symbolic terms into functions

The expression "${{x}^{2}}-1$" is a symbolic term.  You may define a function $f$ whose value at $x$ is given by the expression ${{x}^{2}}-1$. After we say that, "$f$" is a name for the function.

See Functions: images and metaphors.

Naming assertions

You can also give names to symbolic assertions.


Using notation such as “$P(x)$” for statements occurs mostly but not entirely in texts on logic.  (This claim needs lexicographical research.) An overview of its use in first-order logic is given in Mathematical reasoning. See also the Wikipedia articles on various kinds of logic:

Images and metaphors for symbolic expressions

Symbolic terms are encapsulated computations

Algebraic terms are encapsulated computations

A symbolic expression in algebra is both of these things:
$\bullet$ The name of a mathematical object
$\bullet$ An encapsulated computation of the mathematical object it names

If you are fairly proficient in algebra, you already know this subconsciously about algebraic expressions.


Non-algebraic expressions

Most math objects can be combined into new constructions, making expressions like algebraic expressions except that the variables represent structures or objects instead of numbers. Groups, various kinds of spaces, and lots of math objects you never heard of can be combined into "products" and "coproducts", and many of them have "quotients", "function spaces" and other constructions. Most Wikipedia articles about important kinds of math objects describe some of these constructions. The expressions representing such things can still be thought of as both an encapsulated computation and as the name of another math object.

Symbolic expressions as trees

Symbolic expressions such as "$4(x-2)=3$" and the very similar looking "$4x-2=3$" have different abstract structures. The difference results in different solutions: $x=11/4$ and $x=5/4$ respectively. The abstract structures are largely invisible, with the only hint about the difference being the presence or absence of parentheses.

There are other ways to exhibit symbolic expressions that make the abstract structure much more obvious. One way is to use trees. Examples of the tree representation of expressions are given in the following posts in Gyre&Gimble:

I expect to include examples like these in a future revision of this article.

Grammar of the symbolic language

Arrangement of symbols is meaningful

In symbolic expressions, the symbols and the arrangement of the symbols both communicate meaning. 



An expression may contain several subexpressions. The rules for forming expressions and the use of delimiters let you determine the subexpressions.


Math English subexpressions

A phrase in math English can be a subexpression of a symbolic expression.


Embedded symbolic expressions in math English

Symbolic expressions in texts are usually embedded in sentences in math English, although they may stand independently.


Embedded symbolic expressions in math English involves a remarkable number of subtleties. Teachers almost never tell you about these subtleties. The abstractmath article Embedding reveals a few of these secrets. Generally, students learn these facts unconsciously. Some don't, and those generally don't become math majors.


The expression $xy+z$ means $(xy)+z$, not $x(y+z)$.  This is an illustration of the principle that in an algebraic expression, multiplication is performed first, then addition.  We say multiplication has a higher precedence that addition. 


When two operations have the same precedence, the operations should be done from left to right.  The mnemonic “Please Excuse My Dear Aunt Sally” (PEMDAS) describes the order of the common operations:

  1. Parentheses (calculate what is inside the parentheses before you do anything alse.)
  2. Exponentiation
  3. Multiplication and Division
  4. Addition and Subtraction.

One more rule

The names of functions of one variable generally have the highest precedence, except for unary minus, which has lowest precedent.


Irregular syntax in the symbolic language

The symbolic language of math has developed over the centuries the way natural languages do.  In particular, the symbolic language, like English, has definite rules and it has irregularities.


In English, the plural of a noun is normally formed by adding “s” or “es” according to fairly precise rules.  (The plural of car is cars, the plural of loss is losses.)

But English rules have exceptions. Think mouse/mice (instead of mouses) and hold/held (instead of holded for the past tense). .

The symbolic language of math has a lot of rules too. In the symbolic language, the symbol for a function is usually put to the left of the input (argument) and the input is put in parentheses.  For example if $f$ is the function defined by $f(x)=x+1$, then the value of $f$ at $3$ is denoted by $f(3)$ (which of course evaluates to $4$.)  


Just as English has irregular plurals and past tenses, the symbolic language has irregular syntax for certain expressions. Here are two of many examples of irregularities.

There are many other examples of irregularities in symbolic notation in these places:

Other sections of this chapter are in separate files:

Variables and substitution

Variable objects



Other symbols

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