The Pythabacus, Sets and Groups
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  • View Video Demonstrations Below.
  • The bottom row of (gold) beads may be associated with groups of elements from a set. Each count or touch of these beads notes an element in the bead group. Each column in the yellow array is a bead group.
  • Press space bar (PSB) or click up arrow (CUA) to display a subset defining numeral in the upper left corner of dark gray area. Then touch the leftmost bottom gold bead to input subset defining numeral in yellow array. To display numerals click small square under (X) box. Move mouse carefully over beads in order to input more numerals in yellow array.
  • To explore CUA until (0) displays, click leftmost gold bead and click right arrow (CRA) to push beads against right post. CUA until the numeral 1 displays
  • To change subset CUA again. When zero (0) displays no element inputs. Click on element to remove it from array. Click (X) to reset array.
  • Steps 1a: Starting at the leftmost gold bead count three beads (touching each with mouse), click third bead and click left arrow (CLA) to push beads against left post.
  • Step 1b:  Note the upper leftmost spaces in array have inputs of three 1's representing the count of beads.
  • Step 2a: CUA until 2 displays. Starting at the leftmost gold bead against the right post count five beads, click the fifth bead and CLA to push beads to the middle of frame.
  • Step 2b: The  next five spaces will have inputs of five 2's representing the second count of beads.
  • CUA until (0) displays. Click leftmost gold bead of triangle in middle of frame CRA to reveal rectangle of brown beads. As indicated by the arithmetic statements to the left of the frame the count of brown beads in the rectangle is the solution to the demonstrated math operation 3 X 5 = 15.
  • The solution is also equal to the count of unique pairs of elements (1,2) possible between the two subsets articulated in the array. This count is termed the outer product of the unions of the subset groups indicated by the bottom number of the addition statement. Each Brown bead represents a pair of elements.
  • Step 3a: Click the rightmost bead of the rectangle, found on the 6 rod counting from the bottom rod, and CLA to push rectangle against triangle on left post. CUA until the numeral 2 displays again. Then, starting with the gold bead against the left post count after five to seven. CUA until (0) displays. Click the next (third bead of triangle) and CRA to  push it a little to the right.
Video Demonstrations Demonstration Application

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  • Step 3b: The  first two spaces of the second array row will have inputs of 2's representing the continuation of the second count of beads.
  • As indicated by the multiplication arithmetic statements  the solution to the demonstrated math operation is 3 X 7 = 21.
  • The solution though is represented in a different base than 10, because the count of the second number was continued with the leftmost gold bead before all 10 gold beads were included in the counting. Since we only included 8 beads in the counting the base of the solution must be 8. So the two beads of the three on the left post counted to make seven are multiples of the base, 2 X 8 =16 indicated by the top number of the addition statement. The third gold bead separated from the two, with the five gold beads to its right, are the factors of the line of brown beads above the third bead. The count of these brown beads is indicated by the bottom number of the addition statement.
  • Therefore the elements of the set are included in 8 groups (columns in the array).  The count of unique pairs of elements (1,2) found in groups between the subgroups is termed the inner product. These products are the indicated by the red row of numbers above the groups or columns. The sum of these inner products equal the multiple of the base. See this sum at the right end of the red row of numbers (inner products). The outer product equals the count of unique pairs of elements (1,2) found between groups included in the two extensions of  groups that have equal counts of elements. In this case a single group with an element designated by a 1 and five groups with elements designated by 2's.

 

  • Step 4a: Click the topmost bead of the brown line and CLA to push beads back into the triangle against the post.
  • Step 4b: Place mouse over the third (rightmost) bead against the left post and see the input a third 2 in the second row of the array.Now there are three groups with an inner product. Note the sum of inner products is now 3 and 3 times the base 8 equals 24, the top number of the addition statement. There is now no outer product which is indicated by the bottom number being 0.
  • Step 5a: Now CUA until (2) displays. Continue the second count over the five beads in the middle by touching three of the beads. CLA to push beads a little to the left. The brown rectangle will include nine beads indicated by the bottom number of the addition statement. 24 + 9 = 3 X 11 the demontrated math operation.
  • Step 5b: Note there are two extentions of groups that have 2 elements in a group. The groups include the elements (1,2) and (2,2). The count of unique pairs of found between these group produces the outer product of (9). The inner products of the groups with elements (1,2) produce the multiple of the base, giving 3 X 8 = 24.