Editing Napier's bones (section)

== Multiplication ==

The simplest sort of [[multiplication]], a number with multiple digits by a number with a single digit, is done by placing rods representing the multi-digit number in the frame against the left edge. The answer is read off the row corresponding to the single-digit number which is marked on the left of the frame, with a small amount of addition required, as explained in the examples below.

When multiplying a multi-digit number by another multi-digit number, the larger number is set up on the rods in the frame. An intermediate result is produced by the device for multiplication by each of the digits of the smaller number. These are written down and the final result is calculated by pen and paper.

To demonstrate how to use Napier's bones for multiplication, three examples of increasing difficulty are explained below.

===Multiplication by a small single-digit number===

The first example computes {{nowrap|1=425 × 6}}.

Napier's bones for 4, 2, and 5 are placed into the board, in [[sequence]]. These bones show the larger figure which will be multiplied. The numbers lower in each column, or bone, are the digits found by ordinary multiplication tables for the corresponding integer, positioned above and below a diagonal line. (For example, the digits shown in the seventh row of the 4 bone are {{frac|2|8}}, representing {{nowrap|1=7 × 4 = 28}}.) In the example below for {{nowrap|1=425 × 6}}, the bones are here depicted as red (4), yellow (2), and blue (5).
[[File:Napier's Bones ex 1 pic 1.png|center|125px|First step of solving 6 x 425]]
The left-most column, preceding the bones shown coloured, may represent the 1 bone. (A blank space or zero to the upper left of each digit, separated by a diagonal line, should be understood, since {{nowrap|1=1 × 1 = 01}}, {{nowrap|1=1 × 2 = 02}}, {{nowrap|1=1 x 3 = 03}}, etc.) A small number is chosen, usually 2 through 9, by which to multiply the large number. In this example the small number by which to multiply the larger is 6. The horizontal row in which this number stands is the only row needed to perform the remaining calculations and may now be viewed in isolation.
[[File:Napier's Bones ex 1 pic 2.png|center|300px|Second step of solving 6 x 425]]
For the calculation, the digits separated by vertical lines (i.e. paired between diagonal lines, crossing over from one bone to the next) are added together to form the digits of the product. The final (right-most) number on that row will never require addition, as it is always isolated by the last diagonal line, and will always be the final digit of the product. In this example, there are four digits, since there are four groups of bone values lying between diagonal lines. The product's digits will stand in the order as calculated left to right. Apart from the first and the final digit, the product's digits will each be the sum of two values taken from two different bones.
[[File:Napier's Bones ex 1 pic 3.png|center|175px|Third step of solving 6 x 425]]
Bone values are added together, as described above, to find the digits of the product. In this diagram, the third product digit from the yellow and blue bones have their relevant values coloured green. Each sum is written in the space below. The sequence of the summations from left to right produces the figure of 2550. Therefore, the solution to multiplying 425 by 6 is 2550.

===Multiplication by a larger single-digit number===
When multiplying by larger single digits, it is common that upon adding a diagonal column, the sum of the numbers results in a number that is 10 or greater.

The second example computes {{nowrap|6785 × 8}}. 

Like Example 1, the corresponding bones to the biggest number are placed in the board. For this example, bones 6, 7, 8, and 5 were placed in the proper order as shown below.
[[File:Napier's Bones ex 2 pic 1.png|center|175px|First step of solving 6785 × 8]]
In the first column, the number by which the biggest number is multiplied by is located. In this example, the number was 8.  Only row 8 will be used for the remaining calculations, so the rest of the board has been cleared for clarity in explaining the remaining steps.
[[File:Napier's Bones ex 2 pic 2.png|center|375px|Second step of solving 6785 × 8]]
Just as before, each diagonal column is evaluated, starting at the right side. 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 adjacent left column as demonstrated below.
[[File:Napier's Bones ex 2 pic 3.png|center|250px|Third step of solving 6785 × 8]]
After each diagonal column is evaluated, the calculated numbers are read from left to right to produce a final answer; in this example, 54280 was produced. 

Therefore: The solution to multiplying 6785 by 8 is 54280. 

===Multiplication by a multi-digit number===

The third example computes {{nowrap|825 × 913}}.

The corresponding bones to the leading number are placed in the board.  For this example, the bones 8, 2, and 5 were placed in the proper order as shown below.
[[File:Napier's Bones ex 3 pic 1.png|center|125px|First step of solving 825 × 913]]
To multiply by a multi-digit number, multiple rows are reviewed.  For this example, the rows for 9, 1, and 3 have been removed from the board for clarity.
[[File:Napier's Bones ex 3 pic 2.png|center|300px|Second step of solving 825 × 913]]
Each row is evaluated individually and each diagonal column is added as explained in the previous examples. The sums are read from left to right, producing 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.
[[File:Napier's Bones ex 3 pic 3.png|center|600px|Third step of solving 825 × 913]]
Starting with the rightmost digit of the second number, the sums are placed from the rows in sequential order as seen from right to left under each other while utilising a 0 for a place holder.  
    '''2475'''
    '''825'''0
  '''7425'''00

The rows and place holders are [[summation|summed]] to produce a final answer.
     2475
     8250
 + <u>742500</u>
   '''753225'''

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