MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers

Exercise

In this exercise you will construct OR and AND tables for base-10 values 00 through 15 (or, if you are so inclined, through 16) in order to examine and discuss patternings which you uncover in those tables.

A. For each table (OR and AND)

1. Prepare a neat 16 x 16 (17 x 17) grid with generous top and LH margins for the row and column headers.
2. Beginning at the upper left enter the decimal values 00-15 (00-16) and, in a smaller script, their corresponding binary values 0000-1111 (00000 - 10000) both into the row headers and into the column headers.
3. Put the symbol for the proper operator into the box at the intersection of the margins.
4. Into each cell of the grid enter, as appropriate to the table you are constructing, the logical sum (OR) or product (AND), in decimal, of the row and column values for which that cell is the intersection.
   In order to obtain the logical sum or product you will do best to perform the operation on the corresponding binary values, then convert the binary value into decimal.
   Keep an eye out for patterns emerging and for general traits. Check your results -- one check would come from constructing the entire table then, remembering that these are commutative operations, reading one half of the table against the other.
5. When you have completed this stage of the construction of each table (You may find it easier to begin with the OR table.), stop to present at least five significant observations about the table such as:

Unlike arithmetical addition, the logical sum of any two numbers within range m through n (where m is less than n) is never greater than n, e.g 15 + 15 = 30, but 15 OR 15 = 15.

B. After you have completed construction of the two tables and formulated your observations

1. Choose a palette of sixteen (seventeen) widely varying colors in a non-opaque medium.
2. Prepare a legend which assigns a color to each of the values 00 through 15 (16).
3. Color in the squares of the tables accordingly, letting the numerical value 'show through' for verification.
4. Present at least three observations about the ensuing color patterns.

In the evaluation of this exercise, marks will go to those papers which combine neatness, creativity, and well-formulated observations that link with current and recent lecture materials.
You may wish to recopy your tables and observations in a fine version. The first will have the added advantage of helping you perceive patternings.

C. If you constructed a table through 16, you may care to comment, for additional credit, what the additional row and column values may suggest about larger patterns to come and why this might be so.