Extracting Square Roots
Extracting the square root uses an additional bone which looks a bit different from the others as it has three columns on it. The first column has the first nine squares 1, 4, 9, ... 64, 81, the second column has the even numbers 2 through 18, and the last column just has the numbers 1 through 9.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | √ | ||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0/1 | 0/2 | 0/3 | 0/4 | 0/5 | 0/6 | 0/7 | 0/8 | 0/9 | 0/1 2 1 | |
| 2 | 0/2 | 0/4 | 0/6 | 0/8 | 1/0 | 1/2 | 1/4 | 1/6 | 1/8 | 0/4 4 2 | |
| 3 | 0/3 | 0/6 | 0/9 | 1/2 | 1/5 | 1/8 | 2/1 | 2/4 | 2/7 | 0/9 6 3 | |
| 4 | 0/4 | 0/8 | 1/2 | 1/6 | 2/0 | 2/4 | 2/8 | 3/2 | 3/6 | 1/6 8 4 | |
| 5 | 0/5 | 1/0 | 1/5 | 2/0 | 2/5 | 3/0 | 3/5 | 4/0 | 4/5 | 2/5 10 5 | |
| 6 | 0/6 | 1/2 | 1/8 | 2/4 | 3/0 | 3/6 | 4/2 | 4/8 | 5/4 | 3/6 12 6 | |
| 7 | 0/7 | 1/4 | 2/1 | 2/8 | 3/5 | 4/2 | 4/9 | 5/6 | 6/3 | 4/9 14 7 | |
| 8 | 0/8 | 1/6 | 2/4 | 3/2 | 4/0 | 4/8 | 5/6 | 6/4 | 7/2 | 6/4 16 8 | |
| 9 | 0/9 | 1/8 | 2/7 | 3/6 | 4/5 | 5/4 | 6/3 | 7/2 | 8/1 | 8/1 18 9 |
Let's find the square root of 46785399 with the bones.
First, group its digits in twos starting from the right so it looks like this:
- 46 78 53 99
- Note: A number like 85399 would be grouped as 8 53 99
Start with the leftmost group 46. Pick the largest square on the square root bone less than 46, which is 36 from the sixth row.
Because we picked the sixth row, the first digit of the solution is 6.
Now read the second column from the sixth row on the square root bone, 12, and set 12 on the board.
Then subtract the value in the first column of the sixth row, 36, from 46.
Append to this the next group of digits in the number 78, to get the remainder 1078.
At the end of this step, the board and intermediate calculations should look like this:
|
_____________ √46 78 53 99 = 6 -36 -- 10 78 |
Now, "read" the numbers in each row, ignoring the second and third columns from the square root bone and record these. (For example, read the sixth row as : 0/6 1/2 3/6 → 756)
Find the largest number less than the current remainder, 1078. You should find that 1024 from the eighth row is the largest value less than 1078.
|
_____________ √46 78 53 99 = 68 -36 -- 10 78 -10 24 ----- 54 |
As before, append 8 to get the next digit of the square root and subtract the value of the eighth row 1024 from the current remainder 1078 to get 54. Read the second column of the eighth row on the square root bone, 16, and set the number on the board as follows.
The current number on the board is 12. Add to it the first digit of 16, and append the second digit of 16 to the result. So you should set the board to
- 12 + 1 = 13 → append 6 → 136
- Note: If the second column of the square root bone has only one digit, just append it to the current number on board.
The board and intermediate calculations now look like this.
|
_____________ √46 78 53 99 = 68 -36 -- 10 78 -10 24 ----- 54 53 |
Once again, find the row with the largest value less than the current partial remainder 5453. This time, it is the third row with 4089.
|
_____________ √46 78 53 99 = 683 -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 |
The next digit of the square root is 3. Repeat the same steps as before and subtract 4089 from the current remainder 5453 to get 1364 as the next remainder. When you rearrange the board, notice that the second column of the square root bone is 6, a single digit. So just append 6 to the current number on the board 136
- 136 → append 6 → 1366
to set 1366 on the board.
|
_____________ √46 78 53 99 = 683 -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 99 |
Repeat these operations once more. Now the largest value on the board smaller than the current remainder 136499 is 123021 from the ninth row.
In practice, you often don't need to find the value of every row to get the answer. You may be able to guess which row has the answer by looking at the number on the first few bones on the board and comparing it with the first few digits of the remainder. But in these diagrams, we show the values of all rows to make it easier to understand.
As usual, append a 9 to the result and subtract 123021 from the current remainder.
|
_____________ √46 78 53 99 = 6839 -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 99 -12 30 21 -------- 1 34 78 |
You've now "used up" all the digits of our number, and you still have a remainder. This means you've got the integer portion of the square root but there's some fractional bit still left.
Notice that if we've really got the integer part of the square root, the current result squared (6839² = 46771921) must be the largest perfect square smaller than 46785899. Why? The square root of 46785399 is going to be something like 6839.xxxx... This means 6839² is smaller than 46785399, but 6840² is bigger than 46785399—the same thing as saying that 6839² 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 continue to generate more digits of the square root.
Similar to finding the fractional portion of the answer in long division, append two zeros to the remainder to get the new remainder 1347800. The second column of the ninth row of the square root bone is 18 and the current number on the board is 1366. So compute
- 1366 + 1 → 1367 → append 8 → 13678
to set 13678 on the board.
The board and intermediate computations now look like this.
|
_____________ √46 78 53 99 = 6839. -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 99 -12 30 21 -------- 1 34 78 00 |
The ninth row with 1231101 is the largest value smaller than the remainder, so the first digit of the fractional part of the square root is 9.
|
_____________ √46 78 53 99 = 6839.9 -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 99 -12 30 21 -------- 1 34 78 00 -1 23 11 01 ---------- 11 66 99 |
Subtract the value of the ninth row from the remainder and append a couple more zeros to get the new remainder 11669900. The second column on the ninth row is 18 with 13678 on the board, so compute
- 13678 + 1 → 13679 → append 8 → 136798
and set 136798 on the board.
|
_____________ √46 78 53 99 = 6839.9 -36 -- 10 78 -10 24 ----- 54 53 -40 89 ----- 13 64 99 -12 30 21 -------- 1 34 78 00 -1 23 11 01 ---------- 11 66 99 00 |
You can continue 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 reminder becomes zero which means you have the exact square root.
Having found the desired number of digits, you can easily determine whether or not you need to round up; i.e., increment the last digit. You don't need to find another digit to see if it is equal to or greater than five. Simply append 25 to the root and compare that to the remainder; if it is less than or equal to the remainder, then the next digit will be at least five and round up is needed. In the example above, we see that 6839925 is less than 11669900, so we need to round up the root to 6840.0.
There's only one more trick left to describe. If you want to find the square root of a number that isn't an integer, say 54782.917. Everything is the same, except you start out by grouping the digits to the left and right of the decimal point in groups of two.
That is, group 54782.917 as
- 5 47 82 . 91 7
and proceed to extract the square root from these groups of digits.
Read more about this topic: Napier's Bones
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