Written rooting

The Written root extraction is a method for calculating the square root of a rational number , which may be carried out without a computer. It is similar to the written division and provides one digit of the result for each calculation step. The basis of the written root extraction are the binomial formulas .

In schools today, written root extraction is rarely taught, and in earlier times it was only rarely used. The reasons are, on the one hand, the lesser practical importance of root extraction in contrast to the basic arithmetic operations , and on the other hand iterative methods such as the Heron method (Babylonian root extraction) are easier to carry out and usually provide sufficient accuracy more quickly.

The cube root to pull in writing is also possible. This method, which is even more rarely used, is an extension of the principle used for taking the square root. Also roots with higher exponents can be drawn with this method. In addition, all these calculations are also possible in other number systems .

Procedure for the square root

The radical is first divided from the comma to the right and left into groups of two digits each. The first (one or two-digit) group provides the first digit of the result by looking for the largest single-digit number whose square is not larger than this number. The square is subtracted from the first group, the difference is written in the next line and the next group of two of the radicand is added.

For the determination of the next (and each other) the first point comes binomial formula is used: . is the next digit you are looking for , the previous result, for the correct representation with a zero attached. has already been subtracted from the radicand by the previous steps; in order to be able to append the position to the result , the terms and must now be subtracted. ${\ displaystyle (a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2}}$ ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle a ^ {2}}$ ${\ displaystyle b}$ ${\ displaystyle 2ab}$ ${\ displaystyle b ^ {2}}$

The number determined above is divided by the result, but the remainder must not be less than . After subtracting and the next group of two of the radicand is added and the next calculation step is carried out in the same way. The process is ended either when the radicand could be reduced to zero by the repeated subtractions (then the radicand is a square number) or the result is sufficiently precise (any number of zeros can be added as decimal places of the radicand). ${\ displaystyle 2a}$ ${\ displaystyle b}$ ${\ displaystyle b ^ {2}}$ ${\ displaystyle 2ab}$ ${\ displaystyle b ^ {2}}$

Representation using concrete examples

Square root of 2916

The root of 2916 is to be determined:

The first step is to split the sequence of digits into groups of two, starting with the comma. If a comma is missing (as in this example), the starting point is the number on the far right.

 ______
√ 29 16 = ?

The largest square number that is less than or equal to 29 is . The first digit of the result is therefore 5 .. Add the last two digits 16 to the number 4 and you get 416: ${\ displaystyle 25 = 5 \ cdot 5}$ ${\ displaystyle 29-25 = 4}$

 ______
√ 29 16 = 5
 -25
   4 16

To get the second digit of the result (b) you have to divide by (here:) , leaving a sufficient remainder: 416: 100 = 4 with remainder 16. The remainder 16 corresponds to 4², so the calculation goes to zero because 2916 is a square number. ${\ displaystyle 2 \ cdot a}$ ${\ displaystyle 2 \ cdot 50 = 100}$

 ______
√ 29 16 = 54
 -25
  __
   4 16
  -4 00
  -  16
   ____
      0

Similar to the written division, the indented representation is used here to concentrate the calculation on the relevant places.

By merging the bill can be in this process with no trial account to see if the cube root was indeed a perfect square, iterative process on the other hand always provide only an approximation.

The Heron method applied to example 2916 provides the approximation after two iterations if 50 is selected as the starting value . ${\ displaystyle x \ approx 54 {,} 00023}$

If you choose 2916 as the start value, on the other hand, around ten calculation steps have to be carried out for a comparable result.

Square root of 2538413.6976

example

You look for the largest square number that can be subtracted from the first group (in our example 1). Its square root is the first digit of the result.
The square number itself is subtracted from the first group (2 - 1).
The digits of the next group are added to the difference (153).
The last digit of the new number is not taken into account (15) and this is then divided by twice the previous result (15: 2).
The quotient (7) rounded off to an integer is used for the multiplication factors in the next step. The value is added to the divisor (2) and forms the second factor for the multiplication (27 * 7). If the quotient is greater than 9, the number 9 is always used to calculate the factor. If the product is greater than the number from step 3 (153), both factors are reduced by 1 until the number is smaller (27 7 = 189> 153 → 26 6 = 156> 153 → 25 5 = 125 <153).
The last digit of the factor is the next digit of the result (both factors have the same final digit) (5).
The product is now subtracted from the number from step 3. Continue with 3. until the root is extracted.

Extension to higher root exponents and other number systems

If the root exponent is greater than 2, the radicand is not divided into groups of 2, but into groups of length . In addition, the entire calculation can be performed in a place value system with a base other than 10. ${\ displaystyle n}$ ${\ displaystyle n}$

Examples

Square root of 2 binary

      1. 0  1  1  0  1
    ------------------
   / 10.00 00 00 00 00     1
/\/   1                  + 1
     -----               ----
      1 00                100
         0               +  0
     --------            -----
      1 00 00             1001
        10 01            +   1
     -----------         ------
         1 11 00          10101
         1 01 01         +    1
         ----------      -------
            1 11 00       101100
                  0      +     0
            ----------   --------
            1 11 00 00    1011001
            1 01 10 01          1
            ----------
               1 01 11 Rest

Square root of 3

     1. 7  3  2  0  5
    ----------------------
   / 3.00 00 00 00 00
/\/  1 = 20*0*1+1^2
     -
     2 00
     1 89 = 20*1*7+7^2
     ----
       11 00
       10 29 = 20*17*3+3^2
       -----
          71 00
          69 24 = 20*173*2+2^2
          -----
           1 76 00
                 0 = 20*1732*0+0^2
           -------
           1 76 00 00
           1 73 20 25 = 20*17320*5+5^2
           ----------
              2 79 75

Cube root of 5

     1.  7   0   9   9   7
    ----------------------
  3/ 5.000 000 000 000 000
/\/  1 = 300*(0^2)*1+30*0*(1^2)+1^3
     -
     4 000
     3 913 = 300*(1^2)*7+30*1*(7^2)+7^3
     -----
        87 000
             0 = 300*(17^2)*0+30*17*(0^2)+0^3
       -------
        87 000 000
        78 443 829 = 300*(170^2)*9+30*170*(9^2)+9^3
        ----------
         8 556 171 000
         7 889 992 299 = 300*(1709^2)*9+30*1709*(9^2)+9^3
         -------------
           666 178 701 000
           614 014 317 973 = 300*(17099^2)*7+30*17099*(7^2)+7^3
           ---------------
            52 164 383 027

Fourth root of 7

     1.   6    2    6    5    7
    ---------------------------
  4/ 7.
/\/  -
     6 0000
     5 5536 = 4000*(1^3)*6+600*(1^2)*(6^2)+40*1*(6^3)+6^4
     ------
       4464 0000
       3338 7536 = 4000*(16^3)*2+600*(16^2)*(2^2)+40*16*(2^3)+2^4
       ---------
       1125 2464 0000
       1026 0494 3376 = 4000*(162^3)*6+600*(162^2)*(6^2)+40*162*(6^3)+6^4
       --------------
         99 1969 6624 0000
         86 0185 1379 0625 = 4000*(1626^3)*5+600*(1626^2)*(5^2)+
         -----------------   40*1626*(5^3)+5^4
         13 1784 5244 9375 0000
         12 0489 2414 6927 3201 = 4000*(16265^3)*7+600*(16265^2)*(7^2)+
         ----------------------   40*16265*(7^3)+7^4
          1 1295 2830 2447 6799

Web links

Wikisource: Root (mathematical) - Article of the 4th edition of Meyers Konversations-Lexikon

The written drawing of cube roots ( Memento of June 8, 2001 in the Internet Archive )

Written square root extraction Detailed explanation of the written extraction of roots
detailed explanation of the algorithm with online generator