Napier bone is a manually operated counter made by John Napier of Merchiston for product calculations and quotients of numbers. This method is based on Arabic mathematics and lattice multiplication used by Matrakci Nasuh in the works of Umdet-ul Hisab and Fibonacci in his book Liber Abaci. The technique is also called Rabdology. Napier published his version in 1617 in Rabdology, printed in Edinburgh, Scotland, dedicated to his patron Alexander Seton.
Using the multiplication table embedded in the trunk, multiplication can be reduced to additional operations and division into subtraction. The use of a more advanced rod can even extract the square root. Note that the Napier bone is not the same as the logarithm, which is also associated with the name Napier.
Complete devices usually include a basic board with a rim; the user places the Napier rod in the rim to perform multiplication or division. The left edge of the board is divided into 9 squares, holding numbers 1 to 9. Napier's rods consist of heavy wood, metal or cardboard pieces. Napier bones are three-dimensional, square in cross-section, with four different stems engraved on each. A set like bone may be flanked with a comfortable carrying case.
The surface of a rod consists of 9 boxes, and each box, except for the top, consists of two parts divided by a diagonal line. The first box of each bar holds one digit, and the other box holds this number double, triple, quadruple, quintuple, and so on until the last box contains nine times the number in the top box. Each product's digits are written one for each side of the diagonal; the number less than 10 occupies the lower triangle.
Video Napier's bones
Multiplication
To demonstrate how to use Napier's Bones for multiplication, three examples of difficulty are described below.
Example 1
The first example counts 425 x 6.
Start by placing the bone that matches the main number of problems to the board. If the number 0 is used in this number, a space is left between the bones corresponding to where 0 will be. In this example, bones 4, 2, and 5 are placed in the correct order as shown below.
Look at the first column, select the number you want multiplied by. In this example, the number is 6. The line this number is located in is the only line required to perform the remaining calculations and thus the rest of the board is cleaned below to allow more clarity in the remaining steps.
Starting from the right side of the line, evaluate the diagonal columns by adding numbers that share the same diagonal columns. Single numbers are still that number.
After the diagonal columns are evaluated, one must read from left to right a calculated number for each diagonal column. For this example, reading the sum from left to right produces a final answer of 2550.
Therefore: The solution for multiplying 425 by 6 is 2550. (425 x 6 = 2550)
Example 2
When multiplying with larger single digits, it is common that when adding diagonal columns, the number of numbers yields a number that is 10 or greater. The following example shows how to properly carry more than dozens of places when this happens.
The second example counts 6785 x 8.
Start as in Example 1 above and place it on the bone board that matches the main number of problems. For this example, bones 6, 7, 8, and 5 are placed in the proper order as shown below. (Note that line 7 on bone 8 should be read 5/6 instead of 5/4)
In the first column, find the number you want multiplied by. In this example, the number is 8. With only need to use line 8 for the remaining calculations, the rest of the board below has been cleared for clarity in explaining the remaining steps.
Just like before, start on the right side of the line and evaluate each diagonal column. If the number of diagonal columns is equal to 10 or greater, where dozens of these quantities must be brought and added together with the numbers in the diagonal column to the left directly as shown below.
After each diagonal column is evaluated, the calculated number can be read from left to right to produce the final answer. Reading the sum from left to right, in this example, yields the final answer of 54280.
Therefore: The solution to multiply 6785 by 8 is 54280. (6785 x 8 = 54280)
Example 3
The third example counts 825 x 913.
Begin once more by placing the appropriate bone to the front number on the board. For this example bones 8, 2, and 5 are placed in the correct order as shown below.
When the number you want multiplied by contains several digits, some of the rows must be reviewed. For this example, rows for 9, 1, and 3 have been removed from the board, as seen below, for easier evaluation.
Evaluate each row individually, add each diagonal column as described in the previous example. Reading this number from left to right will generate the number needed for the long hand sum calculation to follow. For this example, Lines 9, Row 1, and Row 3 are evaluated separately to produce the results shown below.
For the final step of the solution, start by writing the numbers multiplied by each other, drawing a line below the second digit.
825 x 913
Starting with the rightmost right digit of the second digit, place the results of the rows in sequential order as viewed from right to left below each other while utilizing 0 for the place holder.
825 x 913 Â Â Â 2475 Â Â Â 8250 Â 742500
Lines and place holders can then be summed to produce the final answer.
825 x 913 Â Â Â 2475 Â Â Â 8250 742500 Â 753225
In this example, the final answer generated is 753225.
Therefore: The solution to multiply 825 by 913 is 753225. (825 x 913 = 753225)
Maps Napier's bones
Division
Distribution can be done in the same way. Let's share 46785399 with 96431, the two numbers we used in the previous example. Place the bar for the divider (96431) on the board, as shown in the graph below. Using an abacus, find all the product dividers from 1 to 9 by reading the numbers shown. Notice that dividends have eight digits, while partial products (save for the first) all have six. So you should temporarily ignore the last two digits of 46785399, ie '99 ', leaving the number 467853. Next, find the largest partial product that is less than the truncated dividend. In this case, this is 385724. You should mark two things, as you can see in the diagram: since 385724 is in the '4' abacus, mark '4' as the leftmost digit of the share; also write the partial product, left flat, under the original dividend, and subtract two tribes. You get the difference as 8212999. Repeat the same steps as above: cut numbers up to six digits, select the partial product immediately less than the truncated number, write the line number as the next digit of the quotient, and subtract some of the product from the difference found in the loop first. Following the diagram should clarify this. Repeat this cycle until the resulting reduction is less than the divisor. The remaining amount is the rest.
Jadi dalam contoh ini, kita mendapatkan hasil bagi 485 dengan sisa 16364. Kita bisa berhenti di sini dan menggunakan bentuk pecahan dari jawaban .
If you like, we can also find as many decimal places as we need by continuing the cycle as in the standard length division. Mark the decimal point after the last digit of the quotient and add zero to the rest so we now have 163640. Continue the cycle, but each time add a zero to the result after subtraction.
Let's do a few digits. The first digit after the decimal point is 1, since the largest partial product less than 163640 is 96431, from line 1. Reduce 96431 from 163640, we go with 67209. Adding zero, we have 672090 to be considered for the next cycle (with partial results 485.1 ) The second digit after the decimal point is 6, as the largest partial product less than 672090 is 578586 from line 6. The partial result is now 485.16, and so on.
Extracting the square root
Quadratic root extract uses extra bone which looks slightly different from the others because it has three columns on it. The first column has the first nine boxes 1, 4, 9,... 64, 81, the second column has an even number of 2 to 18, and the last column only has numbers 1 through 9.
Let's find the square root of 46785399 with the bone.
First, group the digits in pairs from the right so it looks like this:
- 46 78 53 99
- Note: A number like 85399 will be bundled as 8 53 99
Start with the left-most group 46. Choose the largest square on the square root square less than 46, which is 36 from the sixth row.
Since we chose the sixth line, the first digit of the solution is 6.
Now read the second column of the sixth row on the square bone root, 12, and set 12 on the board.
Then subtract the value in the first column of the sixth row, 36, out of 46.
Add this to the next digit group at number 78, to get the remainder 1078.
At the end of this step, the middle and board calculations should look like this:
Now, "read" the numbers in each row, ignore the second and third columns of the square root bone and note this. (For example, read the sixth line as: 0 / 6 1 / 2 3 / 6 -> 756)
Find the largest number smaller than the current one, 1078. You should find that 1024 of the eighth row is the greatest value less than 1,078.
As before, add 8 to get the next digit of the square root and subtract the eighth row value 1024 from the remaining 1078 to get 54. Read the second column of the eighth row on the square bone root, 16, and set the number on the board as follows.
The current number on the board is 12. Add to the first digit of 16, and add the second digit from 16 to the result. So you have to set the board for
- 12 1 = 13 -> add 6 -> 136
- Note: If the second column of the square bone root has only one digit, just add it to the current number on the board.
The board and middle calculations now look like this.
Again, find the row with the greatest value less than the current 5453 remaining section. This time, the third line with 4089.
The next digit of the square root is 3. Repeat the same steps as before and subtract 4089 from the remaining 5453 to get 1364 as the next remaining. When you rearrange the board, notice that the second column of the square root bone is 6, one digit. So just add 6 to the current number on board 136
- 136 -> add 6 -> 1366
to set 1366 on the board.
Repeat this operation once again. Now the biggest value on the board is smaller than the remaining 136499 currently is 123021 from the ninth row.
In practice, you often do not need to find the value of each line to get the answer. You might be able to guess which row has the answer by looking at the numbers on the first few bones on the board and comparing them with the first few digits of the rest. But in this diagram, we show the value of all the lines to make it easier to understand.
As always, add 9 to the result and subtract 123021 from the rest of the current.
You have now "used" all the digits of our number, and you still have leftovers. This means you get the integer part of the square root but there are some fractional bits left.
Note that if we actually got the integer part of the square root, the current yield of the squared (6839 ² = 46771921) should be the largest perfect square smaller than 46785899. Why? The square root of 46785399 will be something like 6839.xxxx... This means 6839 ² is smaller than 46785399, but 6840 ² is greater than 46785399 - the same thing is saying that 6839 ² is the largest perfect square smaller than 46785399.
This idea is used later on to understand how the technique works, but for now let's proceed to generate more digits of the square root.
Similar to finding the fractional part of the answer in the long division, add two zeros to the rest to get the new rest 1347800. The second column of the ninth line of the square bone root is 18 and the current number on the board is 1366. So count
- 1366 1 -> 1367 -> add 8 -> 13678
to set 13678 on the board.
The board and intermediate computing now looks like this.
The ninth row with 1231101 is the largest value smaller than the rest, so the first digit of the fractional part of the square root is 9.
Subtract the ninth row value from the rest and add a few more zeros to get the new left 11669900. The second column on the ninth row is 18 with 13678 on the board, so count
- 13678 1 -> 13679 -> add 8 -> 136798
and set 136798 on the board.
You can continue on these steps to find as many digits as you need and you stop when you have the precision you want, or if you find that the rest is zero which means you have the right square root.
Once you find the number of digits you want, you can easily determine whether you need to collect or not; that is, add the last digit. You do not need to look for other digits to see if they are equal to or greater than five. Just add 25 to root and compare with the rest; if less than or equal to the rest, then the next digit will be at least five and collecting is required. In the example above, we see that 6839925 is less than 11669900, so we need to collect root into 6840.0.
There is just one more trick left to explain. If you want to find the square root of a number that is not an integer, say 54782.917. All the same, unless you start by grouping the digits to the left and right of the decimal point in group two.
That is, group 54782.917 as
- 5 47 82. 91 7
and proceed to extract the square root of these number groups.
Diagonal modification
During the nineteenth century, Napier's bones underwent a transformation to make it easier to read. The trunk starts to be made with an angle of about 65 Â ° so that the triangle to be added is aligned vertically. In this case, in each unit bar the bar is to the right and ten (or zero) to the left.
The rods are made in such a way that the vertical and horizontal lines are more visible than the line at which the bar is touched, making the two components of each digit the result much more readable. Thus, in the picture it is immediately clear that:
- 987654321 ÃÆ'â € "5 = 4938271605
The Ruler of Genaille-Lucas
In 1891, Henri Genaille discovered the variation of the Napier bone which came to be known as the ruler of Genaille-Lucas. By representing graphically carry, the user can read the results of a simple multiplication problem directly, without medium mental calculations.
The following example calculates 52749Ã, ÃÆ' â € "4 4 = Ã, 210,996.
Abaca card
In addition to the abacus "bone" described earlier, Napier also makes abacus cards. Both devices are reunited in a work held by the National Archaeological Museum of Spain in Madrid.
The apparatus is a wooden box with bone inlay. At the top it contains a "bone" abacus, and at the bottom is an abacus card. Abacus This card consists of 300 cards stored in 30 drawers. One hundred of these cards are covered with numbers (referred to as "number cards"). The remaining two hundred cards contain small triangular holes, which, when placed on top of a numbered card, allow the user to only see certain numbers. With a position capable of this card, multiplication can be made up to the 100 digit length limit, with another 200-digit number.
In addition, the door of the box contains the first strength of the digits, the coefficients of the first power provisions of the binomial and the numerical data of the regular polyhedra.
It is not known who the author of this article is, or whether it is from Spain or derived from a stranger, though it may have originally belonged to the Spanish Academy of Mathematics (created by Philip II) or was a gift from the Prince of Wales. The only thing that is certain is that it is conserved in the Palace, from where it is forwarded to the National library and then to the National Archaeological Museum, where it is still preserved.
In 1876, the Spanish government sent an apparatus to an exhibition of scientific instruments in Kensington, where it received so much attention that some societies consulted with the Spanish representatives about the origins and use of the apparatus.
See also
- The Ruler of Genaille-Lucas
- Pascal Calculator
- Slideshow rule
References
External links
- Java implementation of Napier bone in various number systems on cut-the-knots
- Napier and other bones and many calculators
- How Napier bone works (interactive simulator)
Source of the article : Wikipedia
0 Comments