Multiply using Napiers rods. 8X21 =, 2X865, and 56X414
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For the first multiplication you will need the rods for 2 and 1 and the index frame. There are slightly different variations of Napier’s Rods. I will assume the rods for 2 and 1 look like this when placed alongside the index rod (multiplier):

INDEX 2 1
1 0/2 0/1
2 0/4 0/2 
3 0/6 0/3
4 0/8 0/4
5 1/0 0/5
6 1/2 0/6
7 1/4 0/7
8 1/6 0/8
9 1/8 0/9

The columns of the table represent rods.

The oblique (/) represents the diagonal of a box in the illustration. We are interested in the row with 8 as the index multiplier (shown in bold). To perform multiplication we start with the rightmost digit in the lower half of the box, which is 8. Then we add the digits sitting across the diagonals: in this case 6+0=6, so we write 6 next to the 8: 68. Finally we move to the digit in the upper half of the rod 2, which is 1, and we write this before the 6 to give us 168. So 8×21=168.

To work out 2×865, we select rods for 8, 6 and 5 and line them up with the index rod or frame as we did with the previous multiplication. But this time we have three number rods instead of two.

To save time and space, I’ll just show the row for index multiplier 2:

INDEX 8 6 5
2 1/6 1/2 1/0

Write the last digit first: 0.

Now add across the diagonal: 2+1=3, so we have 30.

Now add across the next diagonal: 6+1=7, so we have 730.

Finally, we place the 1 in front: 1730. So 2×865=1730.

For the last multiplication we have a bit more work to do. We line up the relevant rods with the index rod or frame as before, but we are going to use two rows instead of one.

INDEX 4 1 4
5 2/0 0/5 2/0
6 2/4 0/6 2/4

Index 5 is the tens digit so we start by writing 0 as the last digit: 0.

Then we start using the rods. The last digit for index 5 is 0: so we have 00.

Add across the diagonals: 5+2=7, so now we have 700.

Add across the next diagonals: 0+0=0, so we have 0700.

And finally we have 2 to make 20700.

We now move to the units (ones) digit which is 6. 

The last digit for this index row is 4, so we write 4 as the last digit.

The next digit is 6+2=8 making 84.

The next digit is 4+0=4 making 484.

Finally we have 2 making 2484.

Add 20700 to 2484 to get 23184. 

So 56×414=23184.

 

by Top Rated User (1.1m points)

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