abstractmath.org 2.0
help with abstract math


Produced by Charles Wells     Revised 2017-01-22

Introduction to this website    website TOC    website index   blog

THE SYMBOLIC LANGUAGE OF MATH

CONTENTS

Symbolic expressions

Reading symbolic expressions

Images and metaphors for symbolic expressions

Grammar of the symbolic language

Variables and substitution

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 assertions play the same role in the symbolic language as assertions do in math English.
A symbolic assertion may contain variables and it may be true for some values of the variables and false for others.

Examples

"$\pi\gt0$" is a symbolic assertion. It is true.
"$\pi=3$" is a symbolic assertion. It is false, but it is nevertheless a symbolic assertion.
"$x\gt0$" is true for $x=42$ and many other numbers and false for $x=-.233$ and many other numbers.
The symbolic assertion "$x^2-x-2=0$" is true for the numbers $x=-1$ and $x=2$, but not for any other number.
"$x^2-y^2=(x-y)(x+y)$" is a symbolic assertion with two variables. It is true for all numbers $x$ and $y$. Such a thing is commonly called an algebraic identity or identity relation.
The assertion "$x^2\geq0$" is true for all real numbers, but not for all complex numbers. So the data type of a variable matters in determining whether an assertion containing the variable is always true or not.
The assertion "$x^2\lt0$" is false for all real numbers.
The assertion "${{x}^{2}}-6x+4y\gt0$" 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 $y = 2$. There is more about symbolic assertions like this one in the section on constraints

Symbolic statements

A symbolic statement is a symbolic assertion without variables.

A symbolic statement is either true or false.
A symbolic statement is regarded as a special case of a symbolic assertion. What makes it special is that it contains no variables.

Examples

$\pi\gt0$ and $3^2=9$ are true symbolic statements.
$\pi\lt0$ and $2+3=6$ are false symbolic statements. Even though false, they are still regarded as symbolic statements.

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.

Examples

The expression “${{3}^{2}}$” is a symbolic term. It is another name for the number $9$.
"$x^2$" is a symbolic term containing a variable $x$. This means the term has variable meaning depending on which value is substituted for $x$. For example, if you set $x=-2$, you get $(-2)^2$, another name for $4$.
${{x}^{2}}-6x+4y$ (mentioned above) is symbolic term with two variables. If you substitute $2$ for $x$ and $3$ for $y$ then the expression denotes the integer $4$.

Variations in terminology

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

Symbolic assertions may be called formulas or predicates.
True assertions may be called propositions or sentences.

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

Many people would refer to “${{\text{H}}_{\text{2}}}\text{O}$” as “the formula for water”, but it is not a formula in sense of logic because it does not make a statement.
In everyday usage “proposition” may mean a statement to be debated, or a proposal for action, but in math logic the meaning is simply a statement.
The name “term” comes from mathematical logic. The expression “${{x}^{3}}\left( {{y}^{2}}-1 \right)$” contains symbolic terms ${{x}^{3}}$ and ${{y}^{2}}-1$, which would in math English be called factors rather than "terms".

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.

Examples

Texts on group theory use the expression "${{\text{S}}_{n}}$" to denote the group of all permutations of an $n$-element set, with composition as operation.
The group ${{\text{S}}_{3}}$ contains six elements. You can write them out and make a table of the group multiplication (described here).
All true statements about ${{\text{S}}_{3}}$ are implied by the symbol.
Operations can be performed on groups, just as they can for numbers. One operation is the product of two groups. In particular one can form the product ${{\text{S}}_{n}}\times\mathbb{Z}/3$. ($\mathbb{Z}/3$ is another group.) This is a symbolic term just like "$a\times b$", but in group theory rather than in elementary algebra.

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.

Examples

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

“$x \lt y$” is an assertion – a complete statement. If $x$ and $y$ have specific real number values, then the statement is either true or false.
“$x+y$” is a term – an expression to be evaluated. For this expression, if you plug in $x=2$ and $y=3$ then it names the number $5$. See encapsulated computation below.
To write "If $x\lt y$, then $x\lt y+1$" is the same same as saying, "If $x$ is less than $y$, then $x$ is less than $y+1$". Not only is it OK to say it, it's true.
To say "If $x+y$ then $x\lt x+y$" is nonsense. The expression "$x+y$" is not a sentence and so can't be a clause after "if". In the same way, it is nonsense to say "If my house, then it is white".

Division and fractions

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

The expression "$m/n$" is a term denoting the number obtained by dividing $m$ by $n$. Thus "$12/3$" denotes $4$ and "$12/5$" denotes the number $2.4$.
The expression "$m|n$" is the assertion that "$m$ divides $n$ with no remainder". So for example "$3|12$", read "$3$ divides $12$" or "$12$ is a multiple of $3$", is a true statement and "$5|12$" is a false statement.

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.

Example

Let $P(x)$ be the expression “$x\gt1$”. In this case, you could write statements such as “$P(3)$ is true” and “$P(1/2)$ is false", as well as more complicated statements such as "For any number $x$, if $P(x)$ then $x\gt0$."
Don’t let this notation mislead you into thinking “$P(3)$” is a number. “$P(3)$” is a statement, namely the statement “$3\gt1$”. Of course, $P$ may be thought of as a function $f:\mathbb{R}\to \{\text{true, false}\}$.

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.

Examples

The expression “$2\cdot 2+3$” is both a name for the number $7$ and a description of a particular calculation that gives $7$.

In the math ed literature, we say that the expression “$2\cdot 2+3$” encapsulates the process of multiplying $2$ by itself and adding $3$. See the article What is the object of the encapsulation of a process?.
The expression “$63/9$” is also a name for the number $7$ and encapsulates a different calculation that results in $7$.
The expression “$7$” is a name for the number $7$.
- It encapsulates a proper-name calculation, because in our culture “$7$” is the default symbolic name of the number.
- A proper-name calculation is like referring to "Henry" in a conversation where those present know which Henry you are talking about.
- In the case of $7$, the context is that we are talking about math, where everyone is supposed to know what the symbol "$7$" means.
The expression “$371$” is our default name for $371$. It is in decimal notation and encapsulates the calculation “$3\cdot 100+7\cdot 10+1$”.
The expression “The largest positive root of ${{x}^{3}}-9{{x}^{2}}+15x-7$” is a name for $7$, but that fact requires a more difficult calculation that $2\cdot 2+3$ or $63/9$. Indeed, you don’t even know that the expression is a correctly formed name of a number until you work out that ${{x}^{3}}-9{{x}^{2}}+15x-7$ has a positive root.
The expression \[3{{x}^{2}}-2x-5\] names a variable number. Like most variable mathematical objects, some statements about it must be said to be neither true nor false. For example, "$3{{x}^{2}}-2x-5\leq12$" is neither true nor false. On the other hand, it is a nice calculus exercise to show that “$3{{x}^{2}}-2x-5\le -6$” is definitely false for any $x$.

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.

Examples

“${{\sin }^{2}}x$” , “$\sin 2x$” and “$2\sin x$” all mean different things.
“$x{{2}^{\sin }}$” is meaningless.
“${{\sin }^{2}}x$” and “${{\left( \sin x \right)}^{2}}$” mean the same thing. If you took a class in precalculus or calculus, you may have had this fact expressed explicitly or you may have learned it by osmosis (see “osmosis theory” in the Handbook.).

Subexpressions

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

Examples

The subexpressions in “${{x}^{2}}$” are “x” and “$2$”.
The subexpressions in “${{(2x+5)}^{3}}$” are "$2$", "$x$", “$2x$”, "$5$", “$2x+5$” and "$3$".

Math English subexpressions

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

Example

The set $\left\{ {{n}^{3}}|n\in \mathrm{}\text{, }n\gt0 \right\}$ could also be written as$\left\{ {{n}^{3}}|n\text{ is a positive integer} \right\}$.

Embedded symbolic expressions in math English

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

Examples

"If $x\lt y$, then $x\lt y+1$." This math English sentence occurred earlier in this chapter.
"The indefinite integral of the function $x^2+1$ is $\frac{x^3}{3}+x+C$, where $C$ is an arbitrary real number."
The statement "$\int (x^2+1)dx=\frac{x^3}{3}+x+C$" could occur in a text by itself as a sentence, but that is uncommon except perhaps in lists.

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.

Precedence

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.

PEMDAS

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:

Parentheses (calculate what is inside the parentheses before you do anything alse.)
Exponentiation
Multiplication and Division
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.

Examples

"$2\cdot 3+5$" means do the multiplication first, then add the five, getting 11, whereas "$2\cdot (3+5)$" means do the addition first, then multiply the result by $2$, getting $16$.
"$4+3^2$" means first calculate $3^2=9$, getting $4+9$, then calculate $4+9$, getting $13$. But $(4+3)^2$ means $7^2$.
The expression "$\sin x+y$" means calculate $\sin x$ and add $y$ to the result. The expression "$\sin(x+y)$" means calculate $x+y$, then take the sine of the result.
$-{{3}^{2}}$ requires you to calculate ${{3}^{2}}$ first, then apply the minus sign, yielding $-9$. On the other hand, ${{(-3)}^{2}}$ yields $9$.
Because so many people new to math misread some of these expressions, I have acquired the habit of putting in theoretically unnecessary parentheses for clarity. So for example I would write $(\sin x)+y$ instead of $\sin x+y$ and $-({{3}^{2}})$ instead of $-{{3}^{2}}$.
There is supposed to be a rule that says that ${{2}^{{{x}^{\,y}}}}$ denotes $2{{\,}^{\left( {{x}^{\,y}} \right)}}$, but this is even more widely unknown, so I always write $2{{\,}^{\left( {{x}^{\,y}} \right)}}$.
But note: ${{({{2}^{x}})}^{\,y}}={{2}^{x\,y}}$, which is not usually equal to $2{{\,}^{\left( {{x}^{\,y}} \right)}}$.
Wikipedia has an excellent detailed description of the precedence rules of algebra.

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.

Rules

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 $4$.)

Irregularities

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.

The symbol “!” for the factorial (MW, Wi) function is put after the argument. For example, $6! = 720$.
The parentheses around the argument of a function are omitted for the trig and log (MW, Wi) functions, so we typically write $\sin x$ instead of $\sin(x)$ and $\log x$ instead of $\log(x)$. More about that here.

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

Notation for the value of a function
Variations in meaning
The Handbook (search for "Irregular syntax").

Other sections of this chapter are in separate files: