User:Mopedmeredith

Multiplication
To demonstrate how to use Napier’s Bones for multiplication, three examples of increasing difficulty are explained below.

Example 1
Problem: Multiply 425 by 6 (425 x 6 = ?)

Start by placing the bones corresponding to the leading number of the problem into the board. If a 0 is used in this number, a space is left between the bones corresponding to where the 0 digit would be. In this example, the bones 4, 2, and 5 are placed in the correct order as shown below. Looking at the first column, choose the number wishing to multiply by. In this example, that number is 6. The row this number is located in is the only row needed to perform the remaining calculations and thus the rest of the board is cleared below to allow more clarity in the remaining steps. Starting at the right side of the row, evaluate the diagonal columns by adding the numbers that share the same diagonal column. Single numbers simply remain that number. Once the diagonal columns have been evaluated, one must simply read from left to right the numbers calculated for each diagonal column. For this example, reading the results of the summations from left to right produces the final answer of 2550. Therefore: The solution to multiplying 425 by 6 is 2550. (425 x 6 = 2550)

Example 2
When multiplying by larger single digits, it is common that upon adding a diagonal column, the sum of the numbers result in a number that is 10 or greater. The following example demonstrates how to properly carry over the tens place when this occurs.

Problem: Multiply 6785 by 8 (6785x8=?)

Begin just as in Example 1 above and place in the board the corresponding bones to the leading number of the problem. For this example, the bones 6, 7, 8, and 5 are placed in the proper order as shown below. In the first column, find the number wishing to multiply by. In this example, that number is 8. With only needing to use the row 8 is located in for the remaining calculations, the rest of the board below has been cleared for clarity in explaining the remaining steps. Just as before, start at the right side of the row and evaluate each diagonal column. If the sum of a diagonal column equals 10 or greater, the tens place of this sum must be carried over and added along with the numbers in the diagonal column to the immediate left as demonstrated below. After each diagonal column has been evaluated, the calculated numbers can be read from left to right to produce a final answer. Reading the results of the summations from left to right, in this example, produces a final answer of 54280. Therefore: The solution to multiplying 6785 by 8 is 54280. (6785 x 8 = 54280)

Example 3
Problem: Multiply 825 by 913 (825 x 913 = ?) Begin once again by placing the corresponding bones to the leading number into the board. For this example the bones 8, 2, and 5 are placed in the proper order as shown below. When the number wishing to multiply by contains multiple digits, multiple rows must be reviewed. For the sake of this example, the rows for 9, 1, and 3 have been removed from the board, as seen below, for easier evaluation. Evaluate each row individually, adding each diagonal column as explained in the previous examples. Reading these sums from left to right will produce the numbers needed for the long hand addition calculations to follow. For this example, Row 9, Row 1, and Row 3 were evaluated separately to produce the results shown below. For the final step of the solution, begin by writing the numbers being multiplied one over the other, drawing a line under the second number. 825 x 913 Starting with the right most digit of the second number, place the results from the rows in sequential order as seen from right to left under each other while utilizing a 0 for place holders. 825 x  913 2475   8250  742500

The rows and place holders can then be summed to produce a final answer. 825 x  913 2475   8250 +742500   753225

In this example, the final answer produced is 753225. Therefore: The solution to multiplying 825 by 913 is 753225. (825 x 913 = 753225)