Prependix A: Math Notation On KCT KCT logo

© 1996 & 1999 by Karl Hahn

Math Notation over the Web

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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.

Parentheses, Brackets, and Order of Operations

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 ). When expressions take more than one line, you will see something like:

       / 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 ç       ÷ + n ú × m
   ë è c + d ø     û
When no parentheses are shown, you should do arithmetic operations in the following precedence: exponentiation first, then multiplication, then division, the addition and subtraction.

Addition and Subtraction

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) and associative (that is (a + b) + c = a + (b + c)).


I will use the symbol * to indicate multiplication, so a * b is 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)  or simply  a(b + c) = ab + ac  using the more traditional notation of just leaving out the multiply symbol and letting it be understood (that is when no operator symbol is shown between two expressions you should assume that the two expressions are multiplied). As of 1999, where a multiply symbol is needed to avoid ambiguity, the "×" will be used exclusively.

Fractions and Division

The forward slash character will be used to indicate division. So a / b reads 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 + d
reads (a + b) divided by (c + d) or (a + b) over (c + d).

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:

The exponential function, usually denoted as:
(where x is the independent variable) will commonly be denoted here as:

Absolute Values

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 ) take that to mean the square root of that expression. A square root is always the positive square root unless the ± symbol precedes it, as in ±sqrt(x). In that case, it means both the positive and negative square roots of x.

For higher order roots, I will use the notation of: ( expression )1/n, which can be read as the nth root of expression.

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

    _    _   _   ____
   Ö2   Ö3  Öx   Ö1000

Inequalities and Relational Symbols

a > b means a greater than b.
a < b means a less than b.
a >= b means a greater than or equal to b.
a <= b means a less than or equal to b.
a <> b means 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  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

Ordered Pairs

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 A X B is the Cartesian product of the two sets. That means it is the set of all ordered pairs that you can make by taking the first element of the ordered pair from A and the second from B.


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  =  24
By 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, b(n, k) is the kth binomial coefficient of the nth degree (where k <= n). The formula is:

   b(n, k)  =  -------------
               k! * (n - k)!

The more traditional nomenclature for binomial coefficients is:

   ænö  =          
   èkø     k!(n-k)!

The Sigma Summation Symbol

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:

      /    Aj
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.

    å  Aj


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:

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:

   ---  =  y"
For higher derivatives I shall use:
  ---  =  y(n)
Both 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:

The above reads, the partial derivative of x with respect to y.

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



The symbology of:

    | f(x) dx
reads: the integral from a to b of f(x) dx

A better rendering of this is:

   ò  f(x) dx
   ô f(x) dx

Logical Implication Symbols

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 2
means that statement 1 logically implies statement 2. In other words, if statement 1 is true, then statement 2 must also be true, but not necessarily vice versa. In still other words, statement 1 is a sufficient condition for statement 2.
  statement 1  <=  statement 2
means that statement 1 is implied by statement 2. In other words, if statement 2 is true, then statement 1 must also be true, but not necessarily vice versa. Or statement 2 is true only if statement 1 is true. In still other words, statement 1 is a necessary condition for statement 2.
  statement 1  <=>  statement 2
means 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: statement 1 is a necessary and sufficient condition for statement 2, or statement 1 if and only if statement 2. The phrase if and only if is often abbreviated as iff.

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

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