Introduction to Digital Information:  
Bits and Black and White Cells

Dr. Todd Rowland
todd_rowland@wolfram.com
Wolfram Science Group
http://www.wolframscience.com/reference/education/

Description:

In this lesson, we introduce the idea of representing information as a string of bits, that is to say, 0's and 1's. We display bits graphically using black and white cells. Some pieces of information can be represented directly as bits, but others are more complicated and their representations are less direct.

Pre- requisites:

None

Objectives:

Materials:

Graph paper

Computers

Further Notes:

From NCTM  http://www.nctm.org/standards/

Background:

A bit is the simplest kind of information.  It is either 0 or 1.  It can be used to answer a yes or no question.

A collection of bits in a definite order is called a string of bits.  A string of bits can represent a wide range of things, practically anything you can think of.  We will see examples of that in this lesson.

Sometimes the representation is direct, like a a sequence where there are only two possibilities at each place.  For instance, in a row of rooms along a corridor the lights can be either on or off.

Other times, the representation is less direct and requires some arbitrary choices.  Once these choices have been made, it is easy to see from some simple examples that we can represent words, graphics and other data.

Later in the  course, we will see that strings of bits can even represent programs.  We will be investigating programs which can be specified by small strings of bits.  These programs will consist of a single instruction, called a rule, repeated over and over again.

As with many things, it is important to try to understand the simplest cases before moving on to more complicated cases.

Later, we will see that even these simple programs can be quite interesting.  Some will be easy to understand, some will be difficult to figure out what they do, while others will be practically impossible.  Even though these rules are simple, or rather because simple rules are easier to implement, they have an enormous potential for applications both in the pure sciences and in technology.

Before we start with the rules, we need to become comfortable with bits and the idea of digital information.  Also, we need to familiarize ourselves with the way we will be visualizing strings of bits.

Basically a string of bits is a sequence of 0's and 1's like {0,1,0,1,0}.  We will represent a 0 as a white cell and a 1 as a black cell.  So {0,1,0,1,0} corresponds to a sequence of cells colored white, black, white, black, and white.

Procedure:

Give the background on digital information and simple rules.

Show that the values 0 and 1 can be represented graphically as white and black cells, respectively.

Graphically represent some examples like {0,1,1,1}

and {1,0,1,0,1} using either a computer projector, or on a blackboard.

Show how to represent information in digital form.

Consider words involving two letters like "add":  Set 0 ="a" and 1="d", then "add" is represented by {0,1,1}, while "dad" is represented by {1,0,1}.  Obviously, this is limited so next we go to words involving four letters.  The thing is that we need to use more than one bit for each letter.  Set {0,0}="a" {0,1}="e" {1,0}="s" {1,1}="t".  Then "east" is {0,1,0,0,1,0,1,1}.  There are a few other words we can make with these letters like "state" is {1,0,1,1,0,0,1,1,0,1}.  (counting plurals, I know of seven more words with these letters).  We can represent graphics using strings of bits.  We will be using a row of white and black cells to represent a string of bits throughout this course.  A row of colored cells where each cell is either red, green, blue or yellow can be represented in a similar way to the four lettered words above.  Set {0,0}=red, {0,1}=green, {1,0}=blue, {1,1}=yellow.

Red is represented by {0,0} which can be represented graphically as :

Questions:

Answer:  Four in either case.  The number of independent possibilities doesn't depend on what is being represented.  There are some ways to get confused about this...

Answer: Each new bit doubles the possibilities.  (2^n).

Answer:  Many things can be so represented, including numbers.   A discussion can be started about how one might try to encode something as a string of bits, by finding what defines it and encoding that as a string of bits.   Things from the realm of philosophy and theology are a matter of opinion or belief.  Other things are in a gray area, which may or may not turn out to be digitally expressible, like human emotions.

Assessment: