Click here for notes on sending math notation over the email.
The Problem Well, the sad truth is, you can't make all the fancy symbols they use in math texts over the web. Here is a list of symbol conventions I'll be using.
Since I wrote this page several years ago, I have discovered
the "symbol" font, which allows many more math symbols to
be rendered on web pages. So I have annotated this page
in brown to show how some of
the clunky symbology shown here will be replaced by more
readable symbols in most of the pages written during or after 1998.
If you are interested in how the math symbols shown in these
annotations are created so that you can use them in your
own web pages, I invite you to save this page to a file,
and then use Wordpad (or some other text editor) to examine it
in source form. You will see by example how I did it.
If you can't seem to get all the wonderful glyphs that
the symbol font offers, click here
and read the browser notes for some tips.
I will use parentheses and backets in the standard way, to specify the
order of operation in arithmetic expressions. I will use parentheses
in preference to brackets for this purpose. When expressions use only
one line of typed text, parentheses will simply show as in
/ a + b \
( ----- ) + n
\ c + d /
æ a + b ö
ç ÷ + n
è c + d ø
Wherever parentheses appear, they indicate that the operations contained
inside the parentheses should be done before those outside of the parentheses.
I shall use brackets primarily to indicate subscripts. This is in line with how computer program languages indicate subscripting. So an expression of A[13] can be read as A sub 13. When it is not confusing, I shall use A13 to mean the same thing. Sometimes (and only when it is unavoidable) I will use brackets to indicate grouping of operations. Typically they will be used for higher groupings than he parentheses. So you might see something like:
-- -- | / a + b \ | | ( ----- ) + n | * m | \ c + d / | -- --
é æ a + b ö ù ê çWhen no parentheses are shown, you should do arithmetic operations in the following precedence: exponentiation first, then multiplication, then division, the addition and subtraction.÷ + n ú × m ë è c + d ø û
I will use the ordinary symbols: + for addtion and -
for subtraction. So a + b means a plus b,
and a - b means a minus b. I assume that
you already know that addition is commutative (that is
I will use the symbol * to indicate multiplication, so
Multiplication is better rendered using the "×" symbol.
The above equation would be
The forward slash character will be used to indicate division. So
a + b ----- c + dreads
Quotients in later pages will be rendered with a solid line:
a + b
c + d
The expression, x^y indicates x raised to the y power. The following superscripting means the same thing:
xyThe exponential function, usually denoted as:
ex(where x is the independent variable) will commonly be denoted here as:
exp(x)
The notation |x| denotes the absolute value of x. That means if x is negative, its absolute value is positive, but of the same magnitude. If x is positive or zero, the its absolute value is the same as x.
You know those fishhook-like symbols that they use in math books to indicate
square roots? Well I can't make those over the web. So whenever you see
For higher order roots, I will use the notation of:
For taking the square root of numerals and sometimes for square roots of
very short expressions, I will be using
_ _ _ ____
Ö2 Ö3 Öx Ö1000
Most variables symbols will be single letters. Some examples are
a, b, i, j, t, u, v, x, y. Sometimes I will use upper case
letters as well for variable symbols. Often the upper case letters
will be used to represent constants.
Variables can represent integers, real
numbers, constants, or functions. When the independent variable of a
function needs
to be shown, it will follow the symbol for the function immediately, but
in parentheses. A function, f, that takes an independent variable,
x, for example, will be shown as f(x), which you can
read as f of x. Functions of more than one
independent variable will show the variables set off by commas:
Some variables in calculus are traditionally shown using Greek letters. Since I can't make these symbols over the web, I will spell out Greek letters, such as alpha, beta, epsilon, theta, phi. Uppercase Greek letters will be spelled out in all caps: SIGMA, PI
When letters are used to represent sets or vectors, I shall render them in bold type: A, B, u, v, chi, PSI
Greek letters are now available as such on later pages.
A B G D E Z H Q I K L M N X O P R S T U F C Y W
a b g d e z h q i k l m n x o p r s t u f c y w
For reference, the phonetic names of the Greek letters are, in the same order as
you see them above:
alpha, beta, gamma, delta, epsilon, zeta, eta, theta, iota, kappa,
lambda, mu, nu, xi, omicron, pi, rho, sigma, tau, upsilon, phi,
chi, psi, omega
The standard notation of (x,y) will be used to indicate an ordered pair, which will often mean it is a point on a Cartesian plane. Likewise (x,y,z) is an ordered triple, and might be used to represent a point in Cartesian 3-dimensional space. This notation can be extended to any number of dimensions.
If A is a set and B is a set,
then
The expression, n!, where n is a counting number, indicates the product of all the counting numbers up to and including n, and is called, n factorial. So for example,
4! = 1 * 2 * 3 * 4 = 24
4! = 1 × 2 × 3 × 4 = 24By definition, 0! is given the value of 1.
Very rarely you will see the expression, n!!. This indicates not the factorial of a factorial, but rather, the product of all the odd counting numbers up to and including n. So
7!! = 1 * 3 * 5 * 7 = 105
7!! = 1 × 3 × 5 × 7 = 105
Using factorial expressions, you can make binomial coefficients.
The expression,
n!
b(n, k) = -------------
k! * (n - k)!
The more traditional nomenclature for binomial coefficients is:
n!
ænö =
èkø k!(n-k)!
To show a summation over a series of indexed variables or expression, I shall use the standard sigma notation, as best as it can be done over the net:
n
---
\
/ Aj
---
j=1
The above reads: the summation from
j equal 1 to n of
A sub j. It is the same as:
A1 + A2 + A3 + ... + An
The sigma (summation) notation can be rendered like this as well.
n
å Aj
j=1
There's no good web symbol for infinity either. I shall use oo for that purpose.
There is now: ¥
The expression:
lim f(x)
x --> 0
reads: the limit as x goes to zero of f(x).
lim f(x)
x > 0
means the same thing.
I shall be using several standard notations for taking derivatives. The "d" notation is:
dy
--
dx
where y is a function of x reads as: the derivative
of y with respect to
x. If y is a function
of some independent variable, then
y' also indicates the derivative of y
with respect
to the independent variable.
Second derivatives can be indicated in either of two ways:
d2y --- = y" dx2For higher derivatives I shall use:
dny --- = y(n) dxnBoth of the above indicate the nth derivative of y.
Partial derivatives are a real problem, since there is nothing that looks even remotely like the standard symbol for that (the standard symbol looks like a backward '6'). The notation that I will use really sucks, but it's the best I could come up with:
@x -- @yThe above reads, the partial derivative of x with respect to y.
A much more readable rendering of that same partial derivative is:
¶y
¶x
The symbology of:
b
/
| f(x) dx
/
a
reads: the integral from
a to b of f(x) dx
A better rendering of this is:
b
ò f(x) dx
a
or
b
ó
ô f(x) dx
õ
a
The symbols for logical implication will always be shown in bold type to distinguish them from similar-looking relational symbols (like less than or equal to).
statement 1 => statement 2means that
statement 1 <= statement 2means that
statement 1 <=> statement 2means that each statement implies the other. If either of them are true, then both must be true. This relationship is called logical equivalence. It can also be worded as:
The following symbols are also available for logical implication: Þ Ü Û