The symbolic language ofmath is a distinctive special-purpose language. Unlike mathematicalEnglish, it is not a variety of English. It has actually its ownrules of grammar that are rather different from those of English. You cancommonly check out expressions in the symbolic language in any kind of math write-up writtenin any type of language.

You are watching: A mathematical language of symbols including variables

This chapter discusses facets of the symbolic language that might reason challenges to newcomers. It is not a methodical advent to the symbolic language. You can uncover more information in the links in The langueras of math.

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

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

## Symbolic expressions

The symbolic language consists of symbolic expressions composed in the waymathematicians traditionally create them.

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

Eincredibly 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 claims somepoint.

A symbolic assertion might contain variables and also it may be true forsome worths of the variables and also 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 kind of other numbers and also false for \$x=-.233\$ and many kind of various other numbers.The symbolic assertion "\$x^2-x-2=0\$" is true for the numbers \$x=-1\$ and \$x=2\$, yet not for any various 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 unique case of a symbolic assertion. What renders it unique is that it consists of no variables.Examples\$pigt0\$ and also \$3^2=9\$ are true symbolic statements.\$pilt0\$ 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 termnames something.

Terms play the exact same rolein the symbolic language that descriptions carry out in math English.

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

### Variations in terminology

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

Allthese words, as well as my usage of “term”,can reason cognitive dissonance:

Many type of people would certainly refer to “\$ extH_ ext2 extO\$”as “the formula for water”, however it is not a formula in feeling of logic bereason it does not makea statement.
 This sort of dispute in between various parts of mathhappens all the time.Neither side is best or wrong. Get supplied to it.

### Non-algebraicexpressions

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

ExamplesAll true statements around \$ extS_3\$ are implied by the symbol.

Each branch of math is pertained to with specific specific kinds of mathematical objects, and eincredibly one of them research studies many kind of different kinds of operations on the objects, expressed (usually) in symbolic notation.

Distinguish in between assertions and terms

An essential obstacle many type of civilization brand-new to algebra have is that they don"t pay attention to the distinction betweeen assertions and terms.

Examples

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

“\$x lt y\$” is an assertion – a complete statement. If \$x\$and also \$y\$ have actually certain genuine number worths, then the statement is either true or false.To create "If \$xlt y\$, then \$xlt y+1\$" is the very same exact 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.Division and also fractions

Two icons supplied in the study of integers are infamous for their confusing similarity.

The expression "\$m/n\$" is a term denoting the number derived by dividing \$m\$ by \$n\$. Thus "\$12/3\$" denotes \$4\$ and "\$12/5\$" denotes the number \$2.4\$.

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

 It is wise to be a little paranoid around whether you really understand a specific type of math notation.
Be patient

When you see a complicated assertion or term you need to bepatient. You should stop and unwindit. 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 role \$f\$ whose value at \$x\$is provided by the expression \$x^2-1\$. After we say that, "\$f\$" is a name for the attribute.

See Functions: images and metaphors.

Naming assertions

You have the right to likewise offer names to symbolic assertions.

Example Let \$P(x)\$ be the expression “\$xgt1\$”. In this case, you might create statements such as “\$P(3)\$ is true” and“\$P(1/2)\$ is false", and also more complex statements such as "For any type of number \$x\$, if \$P(x)\$ then \$xgt0\$."Don’t let this notation mislead you into reasoning “\$P(3)\$”is a number. “\$P(3)\$” is a statement,namely the statement “\$3gt1\$”. Of course, \$P\$ might be believed of as aattribute \$f:mathbbR o \texttrue, false\$.

Using notation such as “\$P(x)\$”for statements occurs greatly however not totally in messages on logic. (This case demands lexicographical research.) An overview of its use in first-order logic is provided in Mathematical reasoning. See also the Wikipedia posts on miscellaneous 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 knowledgeable in algebra, you already knowthis subconsciously about algebraic expressions.

ExamplesThe expression “\$2cdot 2+3\$” is both a namefor the number \$7\$ and also a description of a particular calculation that gives \$7\$. The expression “\$63/9\$” is also a name for the number \$7\$ and encapsulates a various calculation that outcomes in \$7\$.The expression “\$7\$” is a name for the number \$7\$. A proper-name calculation is choose referring to "Henry" in a conversation wbelow those current understand which Henry you are talking around. In the situation of \$7\$, the context is that we are talking around math, wright here everyone is meant to know what the symbol "\$7\$" means.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 needs an extra tough calculation that \$2cdot2+3\$ or \$63/9\$. Certainly, you don’t also understand that the expression is a correctly formedname of a number till you work-related out that \$x^3-9x^2+15x-7\$ has actually a positive root.Non-algebraic expressions

Many math objects can be linked into brand-new constructions, making expressions choose algebraic expressions except that the variables reexisting structures or objects rather of numbers. Groups, assorted kinds of spaces, and numerous math objects you never heard of deserve to be linked into "products" and also "coproducts", and also many type of of them have actually "quotients", "attribute spaces" and also other constructions. Most Wikipedia write-ups around important kinds of math objects explain some of these constructions. The expressions representing such things can still be believed 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 incredibly comparable looking "\$4x-2=3\$" have actually various abstract frameworks. The difference results in different solutions: \$x=11/4\$ and also \$x=5/4\$ respectively. The abstract frameworks are mainly invisible, via the just hint around the difference being the visibility or lack of parentheses.

Tbelow are various other ways to exhilittle bit symbolic expressions that make the abstract structure much more apparent. One way is to usage trees. Examples of the tree depiction of expressions are provided in the complying with write-ups in Gyre&Gimble:

I suppose to include examples like these in a future revision of this post.

## Grammar of the symbolic language

### Arrangement of symbols is meaningful

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

Examples “\$sin ^2x\$” , “\$sin 2x\$” and also “\$2sin x\$” all intend differentpoints. “\$x2^sin \$” is meaningmuch less.

### Subexpressions

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

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

ExampleThe collection \$left inmathrm ext, ngt0 ight\$ can likewise be written as\$left ext is a positive integer ight\$.

### Embedded symbolic expressions in math English

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

Examples"If \$xlt y\$, then \$xlt y+1\$." This math English sentence developed earlier in this chapter."The indefinite integral of the feature \$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\$" might occur in a text by itself as a sentence, however that is unprevalent except possibly in lists.

Embedded symbolic expressions in math English involves a impressive number of subtleties. Teachers almost never before tell you about these subtleties. The 2175forals.com article Embedding reveals a couple of of these tricks. Normally, students learn these facts unconsciously. Some do not, and also those generally don"t end up being math majors.

### Precedence

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

PEMDAS

When two operations have actually the same precedence, the operationsshould be done from left to appropriate. The mnemonic “Please Excusage My Dear AuntSally” (PEMDAS) describes the order of the prevalent operations:

Parentheses (calculate what is inside the parentheses prior to you do anypoint alse.) Exponentiation Multiplication and also Division Addition and Subtraction.One even more rule

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

Instances "\$2cdot 3+5\$" indicates do the multiplication initially, then add the five,gaining 11, whereas "\$2cdot (3+5)\$" means do the enhancement initially, then multiplythe outcome by \$2\$, gaining \$16\$."\$4+3^2\$" suggests first calculate \$3^2=9\$, getting \$4+9\$, then calculate \$4+9\$, acquiring \$13\$. But \$(4+3)^2\$ implies \$7^2\$. The expression "\$sin x+y\$" means calculate \$sin x\$ and add \$y\$ to the result. The expression "\$sin(x+y)\$" suggests calculate \$x+y\$,then take the sine of the outcome. \$-3^2\$ calls for you to calculate \$3^2\$ first, then applythe minus sign, yielding \$-9\$. On the various other hand, \$(-3)^2\$ yields \$9\$. Because so many human being new to math mischeck out some of these expressions, Ihave actually gained the halittle bit of placing in theoretically unessential parentheses forclarity. So for instance I would certainly write \$(sin x)+y\$ rather of \$sin x+y\$and \$-(3^2)\$ instead of \$-3^2\$. Tbelow is supposed to be a dominion that says that \$2^x^,y\$ denotes\$2,^left( x^,y ight)\$, however this is also more widely unrecognized,so I always compose \$2,^left( x^,y ight)\$.But note: \$(2^x)^,y=2^x,y\$,which is not normally equal to \$2,^left( x^,y ight)\$.

### Ircontinual syntaxes in the symbolic language

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

Rules

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

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

The symbolic language of math has actually the majority of rules also.In the symbolic language, the symbol for a function is generally put to the left of theinput (argument) and the input is put inparentheses. For example if \$f\$ is the attribute identified by \$f(x)=x+1\$, then the value of \$f\$ at \$3\$ is deprovided by \$f(3)\$ (which of course evaluates to\$4\$.)

Irregularities

Justas English has actually irconstant plurals and past tenses, thesymbolic language has actually ircontinuous syntax for particular expressions.

See more: The Hammond Postulate States That Reactions Which Are Thermodynamically Endothermic And Kinetically

Here are 2 of many kind of examples of irregularities.

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

Other sections of this chapter are in sepaprice files:

### Other symbols 