An old algorithm for Square Roots

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An old algorithm for Square Roots

The following is an excerpt from ALGEBRA IN EASY STEPS  Third Edition by
Edwin I. Stein D. of Van Nostrand Company 1956 p243. This is the algorithm
that was required in our school district up until about 1970. We learned it,
in most cases including me, not too well. We certainly did know why it worked
or if it would always work. This is a good example of knowing how to
use an algorithm but not understanding the concepts behind it. By the way
the reason it works is based on the fact that one solution of (A+x)^2=B for
small
x relative to A can be approximated by the solution to the much easier equation
A^2+2*a*x=B.

Good Luck
Gary Wardall
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EXERCISE 93 Square Roots

I. Aim: To find the square root of a number.

II. Procedure

1. Separate the number into groups of two figures each, starting at the
decimal point
   and forming the groups, first to the left and then to the right of the
decimal point.
   Note: If there is an odd number of figures to the left of the decimal
point, there will
   be one group containing a single number. However if there is an odd
number of figures
   to the right of the decimal point, add a zero so that each group
contains two figures.

2. Find the largest square which can be subtracted from the first group at
the left.
   Write it under the first group.

3. Write the square root of this largest square above the first group as
the first figure
   of the square root.

4. Subtract the square number from the first group. Annex the next group to
the remainder.

5. Form the trial divisor by multiplying the root already found by 2 and
annexing a zero.
   Note:In the sample solutions the zero is not written but is used mentally.

6. Divide the remainder (step 4) by the trial divisor (step 5). Annex the
quotient to the root
   already found; also add it to the trial divisor to form the complete divisor.

7. Multiply the complete divisor by the new figure of the root.

8. Subtract this product (step 7) from the remainder (step 4).

9. Continue this process until all the groups, have been used or the
desired number of decimal
   places has been obtained.

10. Since each figure of the root is placed directly above its
corresponding group, the decimal
    point in the root is placed directly above the decimal point in the
given number.

11. Check by squaring the root to obtain the given number.

III. Sample Solutions

1. Find the square root of 328,329.

                      5   7   3
                \/  3 2 8 3 2 9
                    2 5
                    ___
                  107)7 8 3
                      7 4 9
                      _____
                 1143)  3 4 2 9
                        3 4 2 9
                        _______
                          ....

                        Answer, 573


2. Find the square root of 935.2 correct to nearest hundredth.


                  3   0.   5   8
                \/9 3 5. 2 0 0 0
                  9
                 ___
                605)3 5 2 0
                    3 0 2 5
                    _______
                6108) 4 9 5 0 0
                      4 8 8 6 4
                      _________
                          6 3 6



Answer, 30.58

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