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Posted 30 June 2017
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.
A symbolic 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 symbolic 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.
Examples
 " p > 0 ". This is a symbolic statement in the symbolic language of mathematics. It is true.
“ p < 0 ” is a false symbolic statement.
2
 The expression x - 6 x + 4 y > 0 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

x 2 - 5x + 4 = 0 is also a symbolic assertion. It is true for x = 1 and x = 4 and false for all other
values of x.
2
 The expression “ 3 ” 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.
2
x ” (which means sin x Чsin x ) is a symbolic term containing a variable x, so it has varia2
ble meaning depending on which value is substituted for x. For example, sin (p / 4) is another
 “ sin
name for 1/2.

x 2 - 6 x + 4 y is symbolic term with two variables. If you substitute 2 for x and 3 for y then the
expression denotes the integer 4.
Reading symbolic expressions
Distinguish between assertions and terms
An expression such as “ xa y ” can be an assertion (it is saying something) or a term (it is naming
something).
 “ x < y” is an assertion – a complete statement. If x and y have specific real number values, then
it is either true or false.
 “ x + y” is a term – an expression to be evaluated.
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.
Note on 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 and true assertions may be
called propositions or sentences. All these words, as well as our use of “term”, can cause cognitive dissonance:
 Many people would refer to “ H2O ” 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
3
(y 2 - 1)” contains sym-
2
bolic terms x and y - 1 , which would in math English be called factors rather than terms.
Giving names to symbolic expressions
2
2
Let f ( x ) = x - 1. This defines a function f(x) whose value at x is given by the expression x - 1 .
We now have a name for the function.
You can also give names to other symbolic expressions that are not functions.
Example
Let P(x) be the expression “x > 1”. In this case, you could write statements such as “P(2.3) is true”
and “P(.8) is false.” Don’t let this notation mislead you into thinking “P(2.3)” is a number. “P(2.3)” is a
statement, namely the statement “2.3 > 1”. Of course, P could be thought of as a function
f : R ® {true, false} . Using notation such as “P(x)” for statements occurs mostly but not entirely in
texts on logic. (This claim needs lexicographical research.)
Images and metaphors for symbolic expressions
A symbolic expression in algebra is both of these things:
® The name of a mathematical object
® An encapsulated computation of the mathematical object it names
If you are fairly proficient in algebra, you already know this implicitly about algebraic expressions,
which most of the examples in this section are.
Examples
 The expression “ 2 Ч2 +
3 ” is both a name for the number 7 and a description of one calcula-
tion that gives 7.
 The expression “63/9” is also a name for the number 7 and a description of a calculation that results in 7.
 The expression “7” is a name for the number 7. It encapsulates the trivial calculation (like pointing at a table and saying “that table”). That’s because in our culture “7” is the default symbolic name
of the number.
 The expression “371” is our default name for 371. It is in decimal notation and encapsulates the
calculation “ 3 Ч100 + 7 Ч10 + 1”.
x 3 - 9 x 2 + 15x - 7 ” is a name for 7, but that fact
requires a more difficult calculation that 2 Ч2 + 3 or 63/9. Indeed, you don’t even know it is a cor The expression “The largest positive root of
rectly formed name of a number until you work out that
 The expression
x 3 - 9 x 2 + 15x - 7 has a positive root.
x 3 - 9 x 2 + 15x - 7 names a variable number. Like most variable mathemati-
cal objects, some statements about it must be said to be neither true nor false. For example,
x 3 - 8x 2 - 31x - 22 Ј 12 is neither true nor false. On the other hand, “ 3 x 2 - 2x - 5 Ј - 6 ” is
definitely false.
Non-algebraic expressions
Symbolic expressions don’t have to have algebraic form and they do not have to name numbers. For
example, texts on group theory use the expression Symn or Sn to denote the group of all permutations
of an n-element set, with composition as operation. Thus
S3 contains 6 elements. You can write them
out and make a table of the group multiplication (described here). I am not sure this should be called an
encapsulated computation, but all those facts about S3 are implied by the symbol.
Grammar of the symbolic language
Symbols and arrangements
In symbolic expressions, the symbols and the arrangement of the symbols both communicate
meaning. For example, as you know,
Examples


2
“ sin
x ” , “ sin2x ” and “ 2sinx ” all mean different things.
sin
“ x2
2
 “ sin
” is meaningless.
2
x ” and “ (sinx) ” mean the same thing: that is one of the rules of understanding symbolic
expressions. If you took a class in trig, 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
2
 The subexpressions in “ x ” are “x” and “2”.
3
 Two of the subexpressions in “ (2 x + 5) ” are “ 2x ” and “ 2x +
5 ”. The rules of algebra require
“ 2x +
5 ” to be enclosed in parentheses, but 2x need not be, although it is not wrong to write
3
“ ((2 x ) + 5) ”.
Math English subexpressions
A phrase in math English can be a subexpression of a symbolic expression.
Example
The set
{n3 | n О N, n > 0}could also be written as {n3 | n is a positive integer}.
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.
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
 Exponentiation
 Multiplication and Division
 Addition and Subtraction.
In addition to that mnemonic, functions of one variable generally have the highest precedence, except for unary minus, which has lowest precedent.
Examples
2 Ч3 + 5 means do the multiplication first, then add the five, getting 11, whereas
2 Ч(3 + 5) means do the addition first, then multiply the result by 2, getting 16.

 The expression sin x + y means calculate sinx and add y to the result. The expression sin(x+y)
means calculate x + y, then take the sine of the result.
- 32 requires you to calculate 32 first, then apply the minus sign, yielding - 9 . On the other
2
hand, (- 3) yields 9.

Remarks
 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 - (32 ) instead of - 32 .
 There is supposed to be a rule that says that 2
unknown, so I always write
2
(x y ) .
xy
denotes
x y
But note: (2 )
x y)
(
, but this is even more widely
2
= 2 x y , which is not usually equal to 2
(x y ) .
 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.)
I don’t even want to think about
tomatoes and avocados.
 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 (mouse/mice, hold/held), the symbolic language
has irregular syntax for certain expressions.
 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). More about that here.
Variables
Substitution
Some symbols used in math