Napier's Chessboard Calculator - Multiplication





 



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Multiplication using a chessboard, as designed by John Napier, is very clever, but very limited in its range. Napier's description shows that he planned to use a working space that was a table much larger than a chessboard, so as to be able to accommodate large number representations.

In this version of the Chessboard Calculator, both the rows and columns of the chessboard are labeled with the binary positional number values. The numbers to be multiplied are initially represented by placing chips on the edge of the board over the positional number values (which can be simulated in this emulation by clicking on the numbers themselves or by entering the desired (decimal) values in the text boxes alongside the edge numbers).

Napier had noted that if one represents the multiplication of two binary position number values as a chip in the cell on the intersection of the row and column on which the two digits lie respectively, then the diagonals represent lines of constant value, as shown in the diagram to the left. Thus once the numbers have been chosen on the edge of the board, the first stage of processing is to place a chip on the cells of the board at the intersections of all rows and columns where numbers have been chosen on the edge. In the emulation this is achieved by clicking READY after the multipliers have been input.

Sliding the chips downwards and diagonally left, keeps the chips on lines of constant value, and the accumulation of chips on the bottom row (corresponding to multiplication by 1) is the addition of the individual chips on the board. This operation is emulated by clicking on SLIDE. Therafter, the coincidence of chips on the bottom row (signified by purple chips) can be eliminated by picking up two chips in a cell and placing a single chip to the cell to the left. (This is emulated by clicking on the purple chips.)

The major limitation of this multiplication calculator is that the result can easily exceed the representation range of the board (maximum 255 decimal). For this reason, the top upper left portion of the board is shaded to show that if a chip is placed in this zone, it would fall off the board when slid diagonally down and left, and thus represents an unrepresentable number!


Last updated 2001/08/09 
© J..A.N. Lee, 2001.