Math Report – John Napier’s Chess Board Calculator
This project is about John Napier’s procedure to perform arithmetic calculations using a chess
board or gird. This method requires each term in the calculation to be broken down into the
sum of binary numbers. Using the binary numbers you can find the sum, difference and product
of many numbers.
This method can also calculate decimal addition as well as the addition of whole numbers. The
answer of the sum of multiple decimal numbers can be performed by separating the equation
into two separate problems such as: 2.4 + 33.33 = (2 + 33) + (0.4 + 0.33), as shown below.
33 + 2 = 35
32 + 1 + 2 = 32 + 2 + 1
32 + 2 + 1 = 35
X
X
X
X
128
64
32
16
8
4
33
2
X
X
2
1
35
33 + 40 = 73
32 + 1 + 32 + 8 = 64 + 8 + 1
64 + 8 + 1= 73
The final answer of the decimal addition is 35.73.
X
X
40
X
X
128
64
X
32
16
8
4
2
X
33
X
73
1
There are certain advantages in expressing numbers in binary notation. Every next increment in
binary notation is double the previous making it easy to perform calculations manually. Another
advantage of having numbers written in binary notation is that if you have two of exactly the
same term and exponent you can easily simplify them one term by just increasing the exponent
by one.
There is two main limitations of using Napier’s chess board to calculate addition, as if you want
to find the sum of two numbers and one of the terms of the two numbers that is in binary
notation is greater than the largest binary number on the x axis, then you have to increase the
grid or chess board to fit larger binary numbers to complete your calculation. There is another
limitation relating to the previous one. If you are finding the sum of numbers with many digits,
you have to have a very large grid or chess board, and this can become very inconvenient but is
still possible.
Using a similar technique to Napier’s addition method you can subtract multiple numbers by
each other. For example:
First you have to put the numbers
into binary notation.
24 – 12
(16+8) – (8+4)
X
X
X
24
X
12
12
128
64
32
16
8
4
2
1
Then you have to make sure a token is
of the larger number is in the same
columns as the smaller one. To do this
you have to divide the token into two
and move right.
XX
XX
24
X
X
12
12
128
64
32
16
8
4
2
1
Now just solve the equation.
((2 x 8) + (2 x 4)) – (8 + 4) = 8 + 4
8 + 4 = 12
128
64
32
16
XX
XX
24
X
X
12
X
X
12
8
4
2
1
The multiplication method produces correct results as proven below:
This follows the simple rule of a regular multiplication table. Each cell contains the product of
both the opposing binary numbers at the end of the row and column. For example the cell “32”
has “8” on its y axis and “4” on its x axis, the product of 8 and 4 equals 32. The tokens are slid
down the board on a diagonal slope towards the left. There is a pattern with each diagonal line
to the bottom left; all of the numbers within each line are the same. The sliding method is
mainly to simplify the answer, as it makes the product in integers multiplied by only one. After
the tokens have been slide down to the bottom left they are again simplified. If there is double
a token it is merged into one and moved to the left, giving you the answer in binary notation
which can be added to find the answer as a single integer.
The multiplication method has a large flaw, as to find the product of two numbers which when
converted into binary notation is very likely to have a term higher than 128 and requires the
person performing this calculation to expand the grid/chessboard to make it large enough to fit
the product.
I have concluded that this method of arithmetic calculations is very powerful and shows the
strength of operating in binary numbers. There are multiple limitations to using this method as
it can be very space consuming and tedious to perform. Over all this would have been a useful
trick for people operating with math in the 16th and 17th centuries.
Addition
X
X
128
64
X
X
32
16
X
X
X
128
64
X
X
8
4
2
X
X
X
X
X
32
16
33
X
15
48
X
X
X
8
4
2
1
79
X
99
X
177
1
You can also add more than two
numbers together.
X
X
X
X
128
64
32
X
X
X
X
X
X
16
8
X
X
21
106
X
25
152
4
2
1
Multiplication
128
64
32
X
X
20 x 5 = 100
4+1 x 16 + 4 = 64 + 32 + 4
4+1 x 16 + 4 = 100
16
8
X
X
4
2
128
X
X
64
32
X
16
8
4
1
2
1
128
64
X
X
32
X
X
16
X
X
8
X
X
4
60 x 3 = 180
4+8+16+32 x 2+1 = 128+32+16+4
4+8+16+32 x 2+1 = 180
2
X
128
64
X
X
32
16
X
8
4
1
2
1
128
64
32
16
X
X
8
4
2
X
128
64
32
X
X
16
8
X
4
2
1
1
9 x 9 x 9 = 81
8+1 x 8+1 = 64+16+1
8+1 x 8+1 = 81