How to Do Long Multiplication Using Napier's Method
What is Napier's method?
Napier's method is a way of calculating long multiplications quickly and easily without the need for a calculator or any complicated arithmetic. It is often called Lattice Multiplication due to the lattice grid used but is commonly known as Napier's method after the Scottish mathematician, physicist and astronomer John Napier (1550-1617), who used the method in his invention of Napier's Bones, which we will look at later.
Despite bearing his name, Napier's method was in common use long before his birth, appearing in mathematical works across Europe, China and Arabia.
Despite bearing his name, Napier's method was in common use long before his birth, appearing in mathematical works across Europe, China and Arabia.
John Napier (1550-1617)
How does Napier's method work?
Napier's method works by splitting the long multiplication up into several single-digit multiplications and additions.
To show this, let's look at the example of 375 × 62.
To start this off we draw a 3 × 2 grid with the 375 on top and the 62 down the right-hand side. We also add diagonal lines running through each square and out to the bottom left. You'll see why these are important in the next step.
To show this, let's look at the example of 375 × 62.
To start this off we draw a 3 × 2 grid with the 375 on top and the 62 down the right-hand side. We also add diagonal lines running through each square and out to the bottom left. You'll see why these are important in the next step.
Completing the multiplication
For each box, we now multiply the number at the top by the number to the right and place the answer in the box so that the units are in the bottom-right of the box and the tens are in the top-left.
For example, the top-left box has a 3 above it and a 6 to the right. 3 × 6 = 18 and so we put the 8 in the bottom-right and the 1 in the top-left.
For example, the top-left box has a 3 above it and a 6 to the right. 3 × 6 = 18 and so we put the 8 in the bottom-right and the 1 in the top-left.
The addition step
For the second step, starting in the bottom-right we look down each diagonal row and add all of the digits in it together. We then write this result in the space at the bottom of the diagonal, outside the box.
In our example, the first diagonal is just the 0 in the bottom-right and so we write 0 underneath. The second diagonal contains 0 + 1 + 4 = 5 and so we write 5 underneath. The third diagonal contains 3 + 2 + 1 + 6 = 12. This answer has a tens value, 1, which we carry to the next diagonal as can be seen in the image below. We write the unit value 2 in our current diagonal. We continue like this until all diagonals have been summed.
In our example, the first diagonal is just the 0 in the bottom-right and so we write 0 underneath. The second diagonal contains 0 + 1 + 4 = 5 and so we write 5 underneath. The third diagonal contains 3 + 2 + 1 + 6 = 12. This answer has a tens value, 1, which we carry to the next diagonal as can be seen in the image below. We write the unit value 2 in our current diagonal. We continue like this until all diagonals have been summed.
Grid with completed addition
The final answer
The final answer can then be found by reading the numbers around the bottom edge starting from the left. We have the numbers 2, 3, 2, 5 and 0 and so 375 × 62 = 23 250.
Even longer multiplication
Napier's method can be extended to any number of digits. If you are multiplying an m-digit number by an n-digit number, simply create an m × n grid and put one number at the top and the other to the right.
In the image below I have completed the multiplication 6128 × 457 using Napier's method. Note how in one diagonal the numbers summed to 20 and so a 2 has been carried to the next diagonal.
By then reading the numbers from the left and bottom, we get that 6128 × 457 = 2 800 496. All of this completed without needing any arithmetic beyond single-digit multiplication and addition.
In the image below I have completed the multiplication 6128 × 457 using Napier's method. Note how in one diagonal the numbers summed to 20 and so a 2 has been carried to the next diagonal.
By then reading the numbers from the left and bottom, we get that 6128 × 457 = 2 800 496. All of this completed without needing any arithmetic beyond single-digit multiplication and addition.
Multiplying 6128 by 457 using Napier's method
Why does Napier's method work?
This method works by automatically collecting together numbers of the same place value.
In the example above we can see that the unit value of the final answer is comprised only of the unit value of 8 × 7 (the final digit of each of the original numbers). The tens value however, is comprised of the tens value of the 8 × 7, plus the unit values from the two occurrences of a ten multiplied by a unit i.e. 2 × 7 and 5 × 8. This happens all the way through until we get to the very end; each diagonal represents one column of place value.
In the example above we can see that the unit value of the final answer is comprised only of the unit value of 8 × 7 (the final digit of each of the original numbers). The tens value however, is comprised of the tens value of the 8 × 7, plus the unit values from the two occurrences of a ten multiplied by a unit i.e. 2 × 7 and 5 × 8. This happens all the way through until we get to the very end; each diagonal represents one column of place value.
What are Napier's bones?
Napier's bones are a set of rods, each rod consisting of a single number at the top (from 0 to 9) and the times table of that number written below in the same format as the method shown above.
For example, here's a picture of the rods for 8, 2 and 5.
For example, here's a picture of the rods for 8, 2 and 5.
Napier's bones for 4, 2 and 5
Mopedmeredith, Wikimedia Commons
How to use Napier's bones?
Napier's Bones use the same mathematics as our method above, but speed it up by providing the answers to the single-digit multiplication already. They work for multiplications involving a one-digit number multiplied by a many-digit number.
In the image above we have the 4, 2 and 5 rods laid out in that order. We can therefore do 7 × 425 by reading out the numbers from the 7 row, making sure to add any digits in a diagonal together first.
Therefore, adding the 8 and 1 in the second diagonal to get 9, and the 4 and 3 in the third column to get 7, we get a final answer of 7 × 425 = 2 975.
To do long multiplication we can use the bones repeatedly with our earlier method to find the answer.
In the image above we have the 4, 2 and 5 rods laid out in that order. We can therefore do 7 × 425 by reading out the numbers from the 7 row, making sure to add any digits in a diagonal together first.
Therefore, adding the 8 and 1 in the second diagonal to get 9, and the 4 and 3 in the third column to get 7, we get a final answer of 7 × 425 = 2 975.
To do long multiplication we can use the bones repeatedly with our earlier method to find the answer.
A set of Napier's Bones used for long multiplication:
Stephencdickson - Wikimedia Commons
Comments
What do you think of Napier's method? Is this a method you use or do you prefer a different method of long multiplication? Don't forget to leave your comments below.