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Posted 23 October
2009
The Symbolic Language of Math
ContentsGrammar of the symbolic language
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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
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©
" ". This is a symbolic statement in the symbolic language of
mathematics. It is true. “
” is a false symbolic statement.
©
The expression 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
©
is also a symbolic assertion. It is true for x = 1 and x = 4 and false
for all other values of x.
©
The expression “ ” is a symbolic term. It is another name for the number
© The expression “<” is a symbolic term. It refers to the less-than relation, which is a mathematical object.
©
“ ” (which means
) is a
symbolic term containing a variable x,
so it has variable
meaning depending on which value is substituted for x.
For example,
is another name for 1/2.
©
is symbolic term with two variables. If
you substitute
Reading
Distinguish between assertions and terms
An expression such as “ ” 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 “ ” 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 “ ” contains symbolic terms
and
,
which would in math English be called factors rather
than terms.
Giving names to symbolic expressions
Let . This defines a function f(x)
whose value at x is given by the
expression
. 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 . 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.
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Examples
©
The expression “ ” is both a name for the number 7 and a description of one calculation 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 “ ”.
©
The
expression “The largest positive root of ” is a name for 7, but that fact requires a
more difficult calculation that
or 63/9. Indeed,
you don’t even know it is a correctly formed name of a number until you work
out that
has a positive root.
©
The
expression names a variable number. Like most variable mathematical objects, some
statements about it must be said to be neither true nor false. For example,
is neither true nor false. On the other hand, “
” 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 or
to denote the
group of all permutations of an n-element set, with composition as operation. Thus
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
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
© “ ” , “
” and “
” all mean different things.
© “ ” is meaningless.
© “ ” and “
” 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
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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 “ ” are “x”
and “
© Two of the subexpressions in “ ” are “
” and “
”. The
rules of algebra require “
” to be enclosed in parentheses, but 2x need not be, although it is not wrong
to write “
”.
Math English subexpressions
A phrase in math English can be a subexpression of a symbolic expression.
Example
The set could also be written as
.
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
© means do the multiplication first, then add
the five, getting 11, whereas
means do the addition first, then multiply the
result by 2, getting 16.
© The expression 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.
© requires you to calculate
first, then apply the minus sign, yielding
. On the other hand,
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 instead of sin x + y and
instead of
.
© There is supposed to be a rule that says that denotes
,
but this is even more widely unknown, so I always write
. But note:
,
which is not usually equal to
.
© 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.)
© 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.