The symbolic language ofmath is a distinctive special-purpose language. Unfavor mathematicalEnglish, it is not a range of English. It has actually its ownrules of grammar that are fairly different from those of English. You cannormally read expressions in the symbolic language in any type of math article writtenin any kind of language.

You are watching: A mathematical language of symbols including variables

This chapter discusses elements of the symbolic language that might reason obstacles to newcomers. It is not a methodical introduction to the symbolic language. You can uncover even more information in the links in The languages of math.

Warning: The terminology offered below to talk about symbolic expressions is nonstandard. See Variations in terminology for more information.

The chapter More aboutthe langueras of math discusses topics that involve both the symboliclanguage and mathematical English.

Symbolic expressions

The symbolic language consists of symbolic expressions written in the waymathematicians traditionally compose them.

A symbolic expression consistsof signs arranged according to specificrules. Every symbolic expression is just one of 2 types: symbolic assertion and also symbolic term.

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

Symbolic assertions

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

A symbolicassertion says somepoint.

A symbolic assertion might contain variables and also it might be true forsome worths of the variables and false for others. Examples"$pigt0$" is a symbolic assertion. It is true."$pi=3$" is a symbolic assertion. It is false, but it is nevertheless a symbolic assertion. "$xgt0$" is true for $x=42$ and also many type of other numbers and false for $x=-.233$ and also many type of various other numbers.The symbolic assertion "$x^2-x-2=0$" is true for the numbers $x=-1$ and also $x=2$, yet not for any kind of other number.The assertion "$x^2lt0$" is false for all real numbers.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 instance of a symbolic assertion. What renders it distinct is that it consists of no variables.Examples$pigt0$ and also $3^2=9$ are true symbolic statements.$pilt0$ and also $2+3=6$ are false symbolic statements. Even though false, they are still concerned as symbolic statements.

Symbolic terms

A symbolic term is a symbolic expression that describes some mathematical object.

A symbolic termnames somepoint.

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

Instances The expression “$3^2$” is a symbolic term. It isan additional name for the number $9$. $x^2-6x+4y$ (stated above) is symbolic term through 2 variables. If you substitute $2$ for $x$ and $3$ for $y$ then the expression denotesthe integer $4$.

Variations in terminology

The names “symbolic assertion” and also “symbolic term” are notstandard consumption in math. In mathematical logic:

Allthese words, as well as my usage of “term”,have the right to cause cognitive dissonance:

Many type of people would certainly refer to “$ extH_ ext2 extO$”as “the formula for water”, yet it is not a formula in sense of logic because it does not makea statement. 

This type of conflict in between different parts of mathhappens all the moment.Neither side is right or wrong. Get provided to it. 


Symbolic expressions don’t have to have actually algebraic develop and theycarry out not need to name numbers.

ExamplesAll true statements about $ extS_3$ are implied by the symbol.

Each branch of math is concerned via particular particular kinds of mathematical objects, and also eexceptionally among them researches many kind of various kinds of operations on the objects, expressed (usually) in symbolic notation.

Reading symbolic expressions

Distinguish in between assertions and terms

A basic challenge many people new to algebra have is that they do not pay attention to the distinction betweeen assertions and also terms.


An expression such as “$xalpha y$, wright here $alpha$ is any kind of old symbol, may be an assertion(saying something) or a term (namingsomething).

“$x lt y$” is an assertion – a finish statement. If $x$and $y$ have specific actual number worths, then the statement is either true or false.To compose "If $xlt y$, then $xlt y+1$" is the exact same same as saying, "If $x$ is much less than $y$, then $x$ is less than $y+1$". Not only is it OK to say it, it"s true.Division and also 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 splitting $m$ by $n$. Therefore "$12/3$" denotes $4$ and "$12/5$" denotes the number $2.4$.

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

It is wise to be a little bit paranoid around whether you really understand a details kind of math notation.

Be patient

When you see a complicated assertion or term you have to bepatient. You need to stop and unwindit. Read the tiresomely long example of unwinding an expression in Zooming and Chunking.

Giving names to symbolic expressions

Turning symbolic terms right into functions

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

See Functions: imperiods and also metaphors.

Naming assertions

You can also provide names to symbolic assertions.

Example Let $P(x)$ be the expression “$xgt1$”. In this situation, you could compose statements such as “$P(3)$ is true” and“$P(1/2)$ is false", and also even more facility statements such as "For any kind of number $x$, if $P(x)$ then $xgt0$."Don’t let this notation mislead you into thinking “$P(3)$”is a number. “$P(3)$” is a statement,namely the statement “$3gt1$”. Of course, $P$ may be believed of as afeature $f:mathbbR o \texttrue, false$.

Using notation such as “$P(x)$”for statements occurs mostly however not totally in messages on logic. (This claim needs lexicographical research.) An overwatch of its usage in first-order logic is given in Mathematical reasoning. See additionally the Wikipedia posts on assorted kinds of logic:

Images andmetaphors for symbolic expressions

Symbolic terms are encapsulated computations

Algebraic terms are encapsulated computations

A symbolic expression in algebra is both of these things:$ullet$ The name of a mathematical object$ullet$ Anencapsulated computation of the mathematical object it names

If you are fairly experienced in algebra, you already knowthis subconsciously about algebraic expressions.

ExamplesThe expression “$2cdot 2+3$” is both a namefor the number $7$ and also a summary of a specific calculation that gives $7$. The expression “$63/9$” is additionally a name for the number $7$ and also encapsulates a various calculation that results in $7$.The expression “$7$” is a name for the number $7$. A proper-name calculation is prefer referring to "Henry" in a conversation where those current recognize which Henry you are talking around. In the situation of $7$, the conmessage is that we are talking around math, where everyone is supposed to know what the symbol "$7$" suggests.The expression “$371$” is our default name for $371$. It is indecimal notation and encapsulates the calculation “$3cdot 100+7cdot 10+1$”. The expression “The largest positive root of $x^3-9x^2+15x-7$”is a name for $7$, but that fact requires a much more challenging calculation that $2cdot2+3$ or $63/9$. Without a doubt, you don’t also know that the expression is a correctly formedname of a number until you occupational out that $x^3-9x^2+15x-7$ has actually a positive root.Non-algebraic expressions

Many math objects can be combined into brand-new constructions, making expressions choose algebraic expressions except that the variables reexisting frameworks or objects rather of numbers. Groups, various kinds of spaces, and also numerous math objects you never heard of can be merged into "products" and "coproducts", and also many type of of them have actually "quotients", "attribute spaces" and also other constructions. Many Wikipedia articles around necessary kinds of math objects define some of these constructions. The expressions representing such points can still be thought of as both an encapsulated computation and as the name of an additional math object.

Symbolic expressions as trees

Symbolic expressions such as "$4(x-2)=3$" and the very equivalent looking "$4x-2=3$" have actually different abstract structures. The distinction results in different solutions: $x=11/4$ and also $x=5/4$ respectively. The abstract frameworks are greatly invisible, via the only hint around the difference being the visibility or absence of parentheses.

Tbelow are other methods to exhibit symbolic expressions that make the abstract framework a lot more obvious. One means is to use trees. Instances of the tree representation of expressions are given in the adhering to posts in Gyre&Gimble:

I mean to incorporate examples prefer these in a future revision of this post.

Grammar of the symbolic language

Arrangement of signs is meaningful

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

Instances “$sin ^2x$” , “$sin 2x$” and also “$2sin x$” all mean differentthings. “$x2^sin $” is meaningless.


An expression may contain numerous subexpressions.The rules for forming expressions and also the use of delimiters let you recognize the subexpressions.

Instances The subexpressions in “$x^2$”are “x” and also “$2$”. The subexpressions in “$(2x+5)^3$”are "$2$", "$x$", “$2x$”, "$5$", “$2x+5$” and also "$3$". Math Englishsubexpressions

Aphrase in math English deserve to be a subexpressionof a symbolic expression.

ExampleThe set $left n^3$ could additionally be written as$left ^3$.

Embedded symbolic expressions in math English

Symbolic expressions in messages are generally embedded in sentences in math English, although they might stand also separately.

Examples"If $xlt y$, then $xlt y+1$." This math English sentence emerged previously in this chapter."The indefinite integral of the attribute $x^2+1$ is $fracx^33+x+C$, where $C$ is an arbitrary real number."The statement "$int (x^2+1)dx=fracx^33+x+C$" could occur in a text by itself as a sentence, however that is unwidespread except perhaps in lists.

Embedded symbolic expressions in math English entails a exceptional number of subtleties. Teachers nearly never tell you about these subtleties. The post Embedding reveals a few of these secrets. Generally, students learn these facts unconsciously. Some do not, and also those mostly don"t come to be math majors.


The expression $xy+z$ suggests $(xy)+z$,not $x(y+z)$. This is an illustration of the principlethat in an algebraic expression, multiplication is performed initially, thenenhancement. We say multiplication has actually a higher precedence that enhancement. 


When two operations have the very same precedence, the operationsneed to be done from left to best. The mnemonic “Please Excusage My Dear AuntSally” (PEMDAS) explains the order of the prevalent operations:

Parentheses (calculate what is inside the parentheses before you execute anything alse.) Exponentiation Multiplication and Division Addition and Subtractivity.One more rule

The names of functions of one variable mainly have actually the greatest precedence,except for unary minus, which has lowestprecedent.

Instances "$2cdot 3+5$" indicates carry out the multiplication first, then add the five,getting 11, whereas "$2cdot (3+5)$" implies do the addition first, then multiplythe outcome by $2$, gaining $16$."$4+3^2$" indicates first calculate $3^2=9$, getting $4+9$, then calculate $4+9$, getting $13$. But $(4+3)^2$ means $7^2$. The expression "$sin x+y$" indicates calculate $sin x$ and also include $y$ to the result. The expression "$sin(x+y)$" implies calculate $x+y$,then take the sine of the outcome. $-3^2$ requires you to calculate $3^2$ initially, then applythe minus authorize, yielding $-9$. On the various other hand, $(-3)^2$ yields $9$. Due to the fact that so many human being brand-new to math misread some of these expressions, Ihave acquired the halittle of placing in theoretically unvital parentheses forclarity. So for example I would compose $(sin x)+y$ rather of $sin x+y$and also $-(3^2)$ instead of $-3^2$. There is supposed to be a rule that says that $2^x^,y$ denotes$2,^left( x^,y ight)$, however this is even even more extensively unwell-known,so I constantly create $2,^left( x^,y ight)$.But note: $(2^x)^,y=2^x,y$,which is not usually equal to $2,^left( x^,y ight)$.

Ircontinuous syntaxation in the symbolic language

The symbolic language of math has actually arisen over thecenturies the way natural languperiods do. In particular, the symbolic language,like English, has actually definite rules and also it has actually irregularities.


In English, the plural of a noun is typically created by including “s” or“es” according to sensibly exact rules. (The plural of auto is cars, the pluralof loss is losses.)

But English rules have actually exceptions. Think mouse/mice (instead of mouses) and also hold/organized (rather of holded for the previous tense). .

The symbolic language of math has actually many rules as well.In the symbolic language, the symbol for a function is typically put to the left of theinput (argument) and also the input is put inparentheses. For instance if $f$ is the feature defined by $f(x)=x+1$, then the value of $f$ at $3$ is delisted by $f(3)$ (which of course evaluates to$4$.)


Justas English has actually irregular plurals and previous tenses, thesymbolic language has actually ircontinuous syntaxes for particular expressions.

See more: ‘ Fifty Shades Of Grey Deleted Scenes, Fifty Shades Freed

Here are two of many examples of irregularities.

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

Other sections of this chapter are in separate files:

Variables and also substitution

Variable objects



Other symbols

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