Typing Help

Typing and the COW

The COW is quite smart, in a bovine sort of way, but she can't read your mind. So when you type in formulas, the COW interprets them very strictly and carefully, according to a prescribed set of rules determined by her programming. The typing rules have been constructed to make the job of entering answers as convenient as possible, but you can still go wrong unless you keep these rules in mind.

Why does it have to be so tricky? Why can't we just type formulas in the same way we would write them on paper? The answer is that ordinary computer keyboards lack two particularly important mathematical operations:

exponentiation
fraction division

In the HTML language, we can represent each of these operations using special formatting tricks:

For exponentiation, use superscripts
x^{y^z}, for example.
For fraction division, use tables

A / B =
A

B

These techniques produce a readable display, but are of no use for entering formulas for mathematical analysis. That's where the typing rules come in: to allow you to enter functions, fractions and exponents in a convenient way.

Here are the most important rules you have to know. For a more detailed discussion, click on Typing Details.

Rule for fractions: The COW has to be told when a series of terms is in the numerator or denominator, so use parentheses to group them. For example:

if you want to enter x+y divided by 2
you type (x+y)/2, not x+y/2
if you want to enter x+y divided by a+b
you type (x+y)/(a+b), not (x+y)/a+b

Rule for exponents: The COW has to be told when an exponent is finished, and the rest of the formula begins. Use parentheses, or a space to terminate the exponent.

For example, suppose you want to type x² / 3

If you type "x^2/3" it will be interpreted as "x^2/3"
If you type "x^2 /3", with a space before the /, it will be interpreted as "x² /3", which is what you want.

In this example, the space terminates the exponent. It tells the COW that after you have typed "x^2", the exponent is finished, and the rest of the formula belongs to the denominator, below the "/". Of course, if you don't mind parentheses, you can type in unambiguously

"(x^2)/3", which just means "(x²)/3", equivalent to what you intend without the parentheses.

More complicated exponential expressions follow the same rule: use a space to terminate the exponent.

Rule for functions: The COW knows about the elementary functions, and allows you to type the argument without parentheses. You can also multiply the argument by constants or variables, but you cannot add without parentheses.

For example:

"sin x" means sin(x)
"ln x" means ln(x)
"cos 2x" means cos(2x)
But "cos x+y" means cos(x)+y, not cos(x+y)

Rule for function powers: The COW understands the usual power notation for functions like the trig and ln functions.

For example:

"sin^2 x" means sin(x)²
"tan^k x" means tan(x)^k
"tan x^k" means tan(x^k)

There are two ways you can improve your facility with this COW notation:

Go to the COW Overview (a button on the COW home page) and click on "Typing answers" to read more, or "Practice typing" to try various formulas.
Within every module is a very useful javascript called "Expression Interpreter". Whenever you type a formula as an answer, you can click on the interpreter link to see how the COW interprets it. If you are entering an answer you think is correct, but the COW doesn't seem to agree, try the interpreter to make sure you are typing the answer you intend to type. In other words, catch your own typographical errors.

That's the basics. Now you have to use the COW and get used to this simplified COW notation. Once you have the idea, it will save you a lot of extra typing time.