Click here for notes on sending math notation over the email.

- The Problem
- Parentheses
- Addition and Subtraction
- Multiplication
- Division
- Exponents
- Absolute Values
- Radicals
- Inequalities and Relational Symbols
- Variables
- The Greek Alphabet
- Ordered Pairs
- Factorial Expressions
- Binomial Coefficients
- The Sigma Summation Symbol
- Infinity
- Limits
- Derivatives
- Partial Derivatives
- Integrals
- Logical Implication Symbols

**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
`( expression )`

/ a + b \ ( ----- ) + n \ c + d /

æ a + b ö çWherever parentheses appear, they indicate that the operations contained inside the parentheses should be done before those outside of the parentheses.~~÷ + n è c + d ø~~

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 `A _{13}` 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
`a + b = b + a``(a + b) + c = a + (b + c)`

I will use the symbol `*` to indicate multiplication, so
`a * b``a` *times* `b`. Sometimes,
when it is clear what is going on, the `*` will be dropped. So
`ab` also reads as `a` *times* `b`. Often, using
fonts and presentation that is available over the net, it becomes difficult
to read when the * is dropped. Hence, I will be including it more often
than some people might like. Just rememeber that the * operator means
multiply. I also assume
that you know that multiplication is distributive over addition and
subtraction -- that is ` a * (b + c) = (a * b) + (a * c)`

Multiplication is better rendered using the "`×`" symbol.
The above equation would be
` a × (b + c) = (a × b) + (a × c) `` a(b + c) = ab + ac ``×`" will be used exclusively.

The forward slash character will be used to indicate division. So
`a / b``a` *divided by* `b` or
`a` *over* `b`. Quite often I will use a horizontal line
made of dashes 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:

xThe exponential function, usually denoted as:^{y}

e(where x is the independent variable) will commonly be denoted here as:^{x}

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
`sqrt( expression )``±` symbol precedes it, as in
`±sqrt(x)``x`.

For higher order roots, I will use the notation of:
`( expression ) ^{1/n}`

For taking the square root of numerals and sometimes for square roots of very short expressions, I will be using

_ _ _ ____ Ö2 Ö3 Öx Ö1000

`a > b``a` *greater than*
`b`.

`a < b``a` *less than* `b`.

`a >= b``a` *greater than or equal to*
`b`.

`a <= b``a` *less than or equal to*
`b`.

`a <> b``a` *not equal to*
`b`.

`a ³ b` means `a` greater than or equal to `b`

`a £ b` means `a` less than or equal to `b`

`a ¹ b` means `a` not equal to `b`

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:
`f(x,y)`

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 wFor reference, the phonetic names of the Greek letters are, in the same order as you see them above:

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

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,

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, `b(n, k)``k`th binomial
coefficient of the `n`th degree (where `k <= n`

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 --- \ / AThe above reads:_{j}--- j=1

A_{1}+ A_{2}+ A_{3}+ ... + A_{n}

The sigma (summation) notation can be rendered like this as well.

n å A_{j}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 --> 0reads:

lim f(x) xmeans the same thing.~~> 0~~

I shall be using several standard notations for taking derivatives. The
"`d`" notation is:

dy -- dxwhere

Second derivatives can be indicated in either of two ways:

dFor higher derivatives I shall use:^{2}y --- = y" dx^{2}

dBoth of the above indicate^{n}y --- = y^{(n)}dx^{n}

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,

A much more readable rendering of that same partial derivative is:

¶y~~¶x~~

The symbology of:

b / | f(x) dx / areads:

A better rendering of this is:

b ò f(x) dx aor

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 1means that=>statement 2

statement 1means that<=statement 2

statement 1means that each statement implies the other. If either of them are true, then both must be true. This relationship is called<=>statement 2

The following symbols are also available for logical implication:
**Þ Ü Û**