Math Notation via Email

© 1998, 2006 by Karl Hahn
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If web page options leave something to be desired with regard to rendering math expressions, email can be a nightmare. But on various math discussion usenet groups, some standard notations seem to have evolved.

Basic Guidelines: First, before you compose any math expressions for sending over the email, you should set the font of your email composition window to an equal-space font like Courier. This will assure that columns will line up the same way for both sender and receiver. Also it is very important to remember to press the enter key before you reach the 80th column of each line. Do not count on the auto-wrap to insert the end-of-lines -- please explicitly press the enter key at the end of each line. Many mail readers render email very poorly if it contains no hard end-of-lines, and your colums will not line up for me the same way as they did for you if you fail to follow this guidline. Please do NOT use special symbols that you cut and paste from web pages or select from font-tables. When they arrive in my email, I cannot read them. And please, do NOT ever use tabs anywhere. If you do you will very nearly assure that columns will not line up for the recipient as they did for you. So always use spaces instead of tabs.

Raising Something to a Power

The notation x^n is almost universally taken to mean xn. It is much easier to do exponentiation using the ^ symbol than to try to line up the exponent in the correct column of the preceding line. If the exponent is an expression, be sure to use parentheses -- x^(n+1) for example. The function, y = ex, you can render either as y = e^x or as y = exp(x). Either notation is widely recognized.

Products

Most of the time you can indicate the product of two things simply by listing them, one after the other, with no operator symbol in between. For example, 2a is taken to mean two times a. There may be cases, though, where this can be hard to read or ambiguous. For example, asin(x) could be taken to mean a times sine of x, or it could be taken to mean arcsine of x. It would be better to write the former as  a sin(x)  or  a*sin(x)  or  a * sin(x).  The * symbol is always taken to mean multiplication unless specified otherwise. If you put the * in to indicate multiplication even where it isn't necessary, there is no harm done.

Square Roots

To render the square root of an expression, you can use sqrt(expression) or (expression)^(1/2). For higher roots, like the cube root, for example, use (expression)^(1/3). Please don't bother trying to construct those fish-hook radical symbols using characters.

Using Parentheses

Use parentheses wherever there is any opportunity for ambiguity.

          ambiguous                unambiguous    unambiguous
                                                             

            1/2a                      1/(2a)        (1/2)a

            2^n-1                     2^(n-1)       (2^n) - 1

            e^x^2                     e^(x^2)       (e^x)^2
Generally, if no parentheses are shown, exponentiation is assumed to be done first, followed by multiplication, division, subtraction, and addition, in that order. Even if you follow that convention, when leaving out the parentheses, it often behooves you to put an additional blank to offset the later-applied operators:
   y  =  x^2  +  a x  +  b
This makes it much easier to read. And if you put in parentheses even where they are redundant, you haven't hurt anything. So when in doubt, put them in.

Do use parentheses when you are using functions:

  y = sin(2x)
Put the parentheses around the argument of the function. If you leave out the parentheses in an example like that, it is especially hard to read. If you want to have, for example, the square of the sine of something, you should use:
  y = sin^2(2x)

Absolute Values

Use |x| to means absolute value of x. You can find the key that produces the | symbol in the upper right of your keyboard, near the backspace key on most standard keyboards. The symbol silk-screened onto the key usually looks like a funny-looking colon. Typing this symbol requires you to use the shift key on the majority of keyboards. You can put any expression you like inside the | symbols.

Derivatives

The f'(x) notation is a lot easier than the

   df
   --
   dx
notation. Use the f'(x) notation wherever feasible. Use the f"(x) for second derivatives. For still higher order derivatives you can use something like f"'(x) for third derivative, f""(x) for fourth, and so on. For very high-order derivatives, the easiest notation to read is
   d^n y
   -----
   dx^n
I will also recognize dy/dx as the derivative of y with respect to x and  d^n y/dx^n  as the nth derivative of y with respect to x.

For partial derivatives, go ahead and use the dy/dx notation, but annotate on the line just before or just after which d-expressions are partials. Or you can do what one of my correspondents has done, and that is to indicate partial derivatives using a <> notation:

   <dx/dt>
means the partial derivative of x with respect to t.

Limits, Summations, and Integrals

Each of these has its own peculiar notation in math books. There is an easy way to write them in email, though, that is self-evident to the reader.

   lim{x->0} x ln(x)
is "the limit as x goes to zero of x times ln(x)."
   integral{0 to r} f(x) dx
is the definite integral taken from 0 to r of f(x) dx. If you leave off the curly-brackets and their contents, you have an indefinite integral. If you like, you can abbreviate the word, "integral" with "int." For double and triple integrals you would have something like
   double integral{r=0 to 1; theta=0 to 2pi} f(r, theta) r dr dtheta
For summations, use, for example
   sum{k=0 to infinity} x^k/k!
You can see that the behavior of the counting variable is described here inside the curly-brackets.

Quotients and Division

No easy way to do these. For small expressions you can use something like 1 / (x^2 + 1). For longer expressions, simply use a row of dashes to make the line, and try to keep your columns aligned properly.

            x^2 - 2x + 1
   y  =  -------------------
         x^3 - 3x^2 + 3x - 1

Subscripting

There are two commonly used ways of doing this. If you are going to be putting expressions into the subscript, use something like

   b[n]  =  n(b[n-1] + b[n-2])
If you are simply going to be using the subscript as a label, for example taking a difference between v-final and v-initial, you can use either the square-bracket notation or something like
   delta v = v_f - v_i

Inequalities and Relational Symbols

Use <= for "less than or equal" and >= for "greater than or equal". You can use either <> or != as a symbol for "not equal". The squiggle, ~, is a good symbol for "approximately equal". Also, << means "much less than", and >> means "much greater than". For the equivalence symbol, , simply use equiv.

Infinity

Please do not send a special symbol like "" or "oo". Instead simply use the word, "infinity," as shown in the example shown above, or you can abbreviate it as "inf."

Greek Letters

You can either spell them out, like delta, epsilon and omega, or you can use the Roman equivalent, which in these cases would be, d, e, and w respectively (use q for theta if you don't want to spell that most popular of Greek letters out). Usually the context will make it clear what you mean.

Vector Operators

If you are sending stuff that involves vectors, you can represent the dot product of two vectors, v and w as

   v dot w
Likewise, their cross product would be
   v cross w
The norm or length of a vector, v, is simply |v|.

If you are dealing with vector fields, use the following equivalents for the "del" operator:

        expression                   email version
                                                  
        f                           grad f
        ∙f                          div f
        ×f                          curl f
        2f                          del^2 f

Special Functions

The expression, [x], is usually used to mean the floor function of x. Another phrase for this function is the greatest integer function. It means, take the greatest integer that is less than or equal to x. So, for example:

   [1.5]      =   1
   [0.99]     =   0
   [-1.1]     =  -2
You can use the [x] notation to represent this, or you can use floor(x).

The function, sgn(x), is 1 for positive values of x-1 for negative values of x,  and zero for  x = 0.

The function, step(x), is 1 for positive values of x,  zero for negative values of x,  and is undefined for  x = 0.

sinh(x)cosh(x)tanh(x)coth(x)sech(x),  and csch(x)  are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant respectively. For the inverse of any of these, just precede its name with arc (e.g., arctanh(x)).


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