Discussion of Different Bases

This topic belongs to middle or even to high school. It can be taught better with a counting board, because it doesn't require inventing new symbols for new digits. Here we use a board with 5 columns, where the last two columns represent a fraction.




Sketch of a Lesson Plan

Students work in groups. Each group is asked to investigate one base B, between 6 and 15, with the counting board shown above.

Each group has to answer the following questions about their base B:
  1. What is the minimal number of tokens needed in one column to represent each number from 1 to B-1, with only white counters, and with white and red counters?
  2. What is the range of all numbers represented in base B on this board? (The answer must have two seperate parts: the range of integers and the range of fractions.)
  3. How do you carry out the algorithm of addition in base B on this board?
Each group presents their answers to the whole class. The presentations may be followed by an open discussion of the possible advantages and disadvantages of each base.


Example in Base 8
  1. What is the minimal number of tokens needed in one column to represent each number from 1 to B-1, with only white counters, and with white and red counters?

    Using only white counters, 2 counters are needed to represent the numbers from 1 to 7. Numbers 1 through 5 can be represented with 1 counter, while numbers 6 and 7 require 2 counters. The same is true when both white and red counters are used.

  2. What is the range of all numbers represented in base B on this board? (The answer must have two seperate parts: the range of integers and the range of fractions.)

    The Range of Integers
    The greatest integer that can be represented on the board is 7778 = 7*82 + 7*81 + 7*80 = 51110.
    The smallest integer than can be represented on the board is -7778 = (-7)*82 + (-7)*81 + (-7)*80 = -51110.
    So, the range of integers is 511 - (-511) = 1022.

    The Range of Fractions
    The greatest fraction that can be represented on the board is 7*(1/8) + 7*(1/64) = 63/6410.
    The smallest fraction that can be represented on the board is (-7)*(1/8) + (-7)*(1/64) = -63/6410.
    So, the range of fractions is 63/64 - (-63/64) = 63/32.

  3. How do you carry out the algorithm of addition in base B on this board?

    4. General Steps
    1. First, represent each number on the same board using white tokens.
    2. Proceeding from right to left, if a column represents a digit greater than N-1, subtract N from the number. Add 1 to the column immediately to the left, and leave the remainder in the original column. Repeat this step until the column holds a digit less than or equal to N-1.
    3. Repeat step 2 for each column until all columns hold digits less than or equal to N-1.

    Example:62 + 7.4


    To begin, the tokens representing 628 and 7.48 are held on the board.



    1 is added to the second column from the left, and 1 remains in the center column.



    At this point, each column holds a digit less than or equal to 7. After regrouping for clarity, the board holds 71.48 = 7*81 + 1*80 + 4*(1/8) = 5710.


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