1 The Blackboard Editor

If your expressions have nothing more complicated than superscripts and subscripts, e.g.,

\[ d_i = | x_i - x_0 | \]

you don’t need to leave the Blackboard editor.

Just select the superscript or subscript text and use the super/subscript buttons to raise or lower the text. This works fine with things like x² or x_i.

It’s not as good when you need both superscripts and subscripts on the same expression. You can get x²_i or x_i² this way, but neither looks quite as good as $x^2_i$.

2 HTML Entities

Sometimes you just need a special character or two. Beware of using your PC operating system’s character insertion utility. It may insert characters that look fine in your operating system, but are invisible or look completely different to people viewing your work from other operating systems and/or other web browsers.

Instead, you can use the HTML button to temporarily view and edit the raw HTML of your text, then type an HTML enitity. An entity in HTML is a named character and is written with a leading ampersand (‘&’) and terminated with a semi-colon (‘;’).

Some useful entities in this course:

Blackboard error warning: the Blackboard spell check facility is not your friend here. Basically, whenever it thinks you have misspelled something, it surrounds the “misspelled” word with the HTML element <span>. Even if you later correct the spelling, the <span>s remain. This can make the underlying HTML so ugly that it is all but impossible to edit.

It also breaks the LaTeX techniques covered later in this document.

For that reason, the spell checker will be turned off throughout this course’s Blackboard area.

3 The Math Editor

When you need more than a character or two, Blackboard has a built-in equation editor, similar to the one found in most word processors. Click the f_x button to bring up a a window similar to the one shown on the right.

Most math is actually a mixture of actual mathematics expressions and plain text explaining/discussing those expressions.

Never use the equation editor for plain text. Get in and out as you move back and forth from math to text. Typesetting plain text as if it were mathematics $often leads to a real mess$, because in mathematics, two characters next to each other represents a product of two variables (even if there is a space between them) and a math editor will typeset them accordingly.

4 LaTeX Mathematics

If you have a lot of math, particularly a derivation or a proof with multiple lines of mathematics that change just slightly from line to line, you may find the Blackboard equation editor awkward because copy and pasting, then making slight alterations to the pasted expression, can be awkward.

A better option, once you get used to it, is to type your expression in TeX format between $$ and $$ markers. For example, $$ \sum_{i=0}^n i $$ typesets as \(\displaystyle\sum_{i=0}^n i\).

You can easily preview your math expression by placing the cursor anywhere inside it, clicking on $f_x$ button, then canceling the preview box that pops up.

4.1 TeX Fundamentals

4.1.1 Using LaTeX in Blackboard

In Blackboard, mathematics may be typeset by placing a TeX expression between $$ and $$. For example $$ x+y$$ yields $x + y$.

4.1.2 Special Characters

There are six special characters in TeX math expressions:

1. ^ is used to indicate a superscript. For example x^2 becomes $x^2$.
2. _ is used to indicate a subscript. For example x_i becomes $x_i$.

   • You might think, “Well, I can do that in HTML or in just about any rich text editor.” But consider the combination of the two, quite common in mathematics, such as x_i^2, which renders as $x_i^2$, not as x_i².

3. The curly brackets { and } are used for grouping mathematical expressions without introducing parentheses or other visible markers.

   Look at the difference between x_i+1, rendering as $x_i+1$, and x_{i+1}, rendering as $x_{i+1}$.

4. The % is used to indicate that the rest of the line is a comment. For example

   \[ x + y = y + x % addition is commutative
   \]
   

   renders as

   \[ x + y = y + x % addition is commutative \]

5. Finally, the backslash \ indicates that characters are losing their normal “meaning” and taking on something new.

   For example, the square brackets are perfectly useful characters: [1, 3, 5] rendering as $[1 ,3, 5]$, but with a backslash in front, they introduce a line of display mode math: \[1, 3, 5\] rendering as \[1 ,3, 5\].

   Similarly, the “normal” meaning of { and } is grouping, as shown above. But if we want to talk about sets, we can add a backslash: $$ \{ x_{i,j} \} $$ renders as $\{ x_{i,j} \}$.

   The most common use of \ will be to introduce “commands” for math that can’t be represented with a single character. This is often done together with { } grouping. For example, $\sqrt{n}$ is typed as \sqrt{n}.

4.1.3 Basic Math

The arithmetic operators function much as you would expect: a * b + c / (d - e) renders as $a * b + c / (d - e)$.

One refinement is available for division. The \frac command typesets division in a vertical style:

\frac{a * b + c}{(d - e)} renders as $\frac{a * b + c}{(d - e)}$.

(Again, take note of the way that { } serve to group expressions together without contributing any visible characters to the final expression.)

4.2 Practice Exercises

4.2.1 Boolean Operators

4.2.2 Relational Operators

A special note about the relational operators. If you are typing math into a program designed to embed mathematics into HTML web pages, you may face a problem due to the face that “<” and “>” are reserved characters in HTML. In that case, you may find that you need to type them like this: x < y and x > y.

4.3 Making It Pretty

4.3.1 Making Things Bigger

Math symbols, particularly operators, can vary considerably in height. TeX has several “resizable” operators that not only tend to be taller than most characters, they can grow and shrink depending on context.

The most common of these are summation and product:

When you start writing large operators and vertical fractions into longer expression, the parentheses and grouping characters can start to look a little wimpy. For example

\[ \sum_{i=0}^n (x_i + i) = (\sum_{i=0}^n x_i) + (\sum_{i=0}^n i) = (n \bar{x}) + (\frac{n(n+1)}{2}) \]  

\[ \sum_{i=0}^n (x_i + i) = (\sum_{i=0}^n x_i) + (\sum_{i=0}^n i) = (n \bar{x}) + (\frac{n(n+1)}{2}) \]

The parentheses in the above example look pretty awful. What we have is mathematically correct, but it’s not pretty.

You can request resizable parentheses or brackets by putting \left and \right in front of them.

\[ \sum_{i=0}^n (x_i + i) = \left(\sum_{i=0}^n x_i \right) + \left(\sum_{i=0}^n i\right) = (n \bar{x}) + \left(\frac{n(n+1)}{2}\right) \]  

\[ \sum_{i=0}^n (x_i + i) = \left(\sum_{i=0}^n x_i \right) + \left(\sum_{i=0}^n i\right) = (n \bar{x}) + \left(\frac{n(n+1)}{2}\right) \]

which certainly looks a lot better.

Every \left should be followed by a \right, though the brackets don’t have to match. \left(0, 10\right], rendering as $\left(0, 10\right]$, is perfectly legal.

If you want to have a left or right bracket that appears to be unmatched, you can use ‘.’ as the bracket character.

That’s how we get expressions like

\[ i(n) = \left\{ \begin{array}{ll} n/2 & \mbox{if n is even} \\
(-n-1) & \mbox{if n is odd} \\ \end{array} \right. \]

but we need a few more tricks before we can write all of that.

4.3.2 Text inside Mathematics

Let’s look at that line “n/2 if n is even” all by itself. If we do that as math:

\[ n/2 if n is even \]

we get a mess:

\[ n/2 if n is even \]

That’s because, normally, when we write characters next to one another, we mean them to represent the product of those single-character variables. Even just the word “if”, entered in math as $$ if$$, will be rendered as $if$, because TeX assumes that by “if” we really mean “$i$ multiplied by $f$”.

Even when you put spaces inside your expression, the typesetter swallows those up in order to show what it thinks you meant by that expression.

If we need to put normal text words inside mathematics, there’s a special command, \mbox, used to indicate that something is actually text.

\[ n/2 \mbox{ if n is even} \]

\[ n/2 \mbox{ if n is even} \]

4.3.3 Arrays

You can arrange things into array-like structures with the array environment:

• Start with \begin{array} and end with \end{array}.
• Just after \begin{array}, you indicate the number of coluns you need and whether the elements in each column should be left-justified, centered, or right-justified by a list of ‘l’, ‘c’, and ‘r’ characters.
• In between the \begin{array} and end with \end{array}, type your entries, separating entries within the same row by & and separatign the rows with \\.

So, for example,

\[ \begin{array}{ll}
    a & b \\
    c & d \\
   \end{array} \]

produces

\[ \begin{array}{ll} a & b \\ c & d \\ \end{array} \]

If that array looks a bit naked, we can give it some surrounding structure using the resizable bracketing we saw earlier:

\[ \left| 
      \begin{array}{ll}
      a & b \\
      c & d \\
     \end{array}
  \right| \]

produces

\[ \left| \begin{array}{ll} a & b \\ c & d \\ \end{array} \right| \]

and

\[ \left[ 
      \begin{array}{ll}
      a & b \\
      c & d \\
     \end{array}
  \right] \]

produces

\[ \left[ \begin{array}{ll} a & b \\ c & d \\ \end{array} \right] \]

Returning to an earlier example

\[ \begin{array}{ll}
    n/2 & \mbox{if n is even} \\
    (-n-1) & \mbox{if n is odd} \\
   \end{array} \]

yields

\[ \begin{array}{ll} n/2 & \mbox{if n is even} \\ (-n-1) & \mbox{if n is odd} \\ \end{array} \]

We can then put a large { on the left and a large “nothing” on the right:

\[ left\\{ \begin{array}{ll}
    n/2 & \mbox{if n is even} \\
    (-n-1) & \mbox{if n is odd} \\
   \end{array} \right. \]

yielding

\[ \left\{ \begin{array}{ll} n/2 & \mbox{if n is even} \\ (-n-1) & \mbox{if n is odd} \\ \end{array} \right. \]

And, finally, we can produce that full conditional expression I mentioned earlier:

\[ i(n) = \left\{ \begin{array}{ll}
    n/2 & \mbox{if n is even} \\
    (-n-1) & \mbox{if n is odd} \\
   \end{array} \right. \]

yielding

\[ i(n) = \left\{ \begin{array}{ll} n/2 & \mbox{if n is even} \\ (-n-1) & \mbox{if n is odd} \\ \end{array} \right. \]

4.3.4 Dots

An ellipsis is a mark used to show that something has been omitted. The most common form of ellipsis is denoted by three periods arranged horizontally. In TeX, this is formed by \ldots:

The natural numbers are $$ 0, 1, 2, \ldots$$ 

renders as “The natural numbers are $0, 1, 2, \ldots$”.

Within arrays, however, we can use \vdots and \ddots to suggest missing elements in a vertical or a diagonal (combined horizontal and vertical) direction.

\[ \left[\begin{array}{llll}
  1      & 0      & \ldots & 0 \\
  0      & 1      & \ldots & 0 \\
  \vdots & \vdots & \ddots & \vdots \\
  0      & 0      & \ldots & 1 \\
\end{array}\right] \]

renders as

\[ \left[\begin{array}{llll} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 \\ \end{array}\right] \]

4.4 Try It Out