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How To Compute The Square Root

The way to compute a square root of a number is more complicated than performing long division.

The only way to tell you how is to walk you through an example so you can see how this is going to work.

We're going to find the square root of 531,441, which is 729, a whole number and rational.

Let's begin by writing the square root of the number by separating it in groups of two like this:

                                 .  
                        /-----------------
                      \/ 53 14 41.00 00 00
You may add the pairs of zeros after the decimal in case the number turns out to be irrational.

What's the biggest single number that when squared can be subtracted from 53? 7 squared is 49, so 7 is the answer. Write the 7 up like this:

                          7      .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49
Add the 7's together on the left column and subtract the 49 from the 53 in the radical like this...
                          7      .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49
   --                    -----
   14                     4
Bring down the "14" from the radicand. The new remainder so far is 414.
                          7      .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49 ..
   --                    -----
   14                     4 14
Here's where it gets tricky. We're going to need to find the second digit of the answer. The rules? You find a single digit "n", add it to 10 times the "14" on the left, and multiply it by "n" to get a number as close as you can get to the remainder without going over.

(10x14 + n) x n
(10x14 + 1) x 1 = 141
(10x14 + 2) x 2 = 284
(10x14 + 3) x 3 = 429

You can't take 429 from 414, 429 is bigger than 414, so the digit is not 3, so you look at digit 2, where 284 can be taken from 414.

Write up the 2's like this:

                          7  2   .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49 .. ..
   --                    -----
   142                    4 14 ..
    +2     142 x 2   =   -2 84 ..
Add the 142 and 2 from the left column and subtract the 284 from the 414 under the radical...
                          7  2   .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49 .. ..
   --                    -----
   142                    4 14 ..
    +2     142 x 2   =   -2 84 ..
   ---                    -------
   144                    1 30
Bring down the "41" from the radicand like this for the new remainder...
                          7  2   .
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49 .. ..
   --                    -----
   142                    4 14 ..
    +2     142 x 2       -2 84 ..
   ---                    -------
   144                    1 30 41
This gets even harder. We're going to need to find the third digit of the answer. The rules again? You find a single digit "n", add it to 10 times the "144" on the left, and multiply it by "n" to get a number as close as you can get to the remainder without going over.

(10x144 + n) x n
(10x144 + 8) x 8 = 11584
(10x144 + 9) x 9 = 13041

13041 matches the remainder, so "9" is the final digit of the answer.

Write up the 9's like this:

                          7  2  9.
                        /-------------------
    7                 \/ 53 14 41.00 00 00
   +7        7 x 7   =  -49 .. ..
   --                    -----
   142                    4 14 ..
    +2     142 x 2   =   -2 84 ..
   ---                    -------
   1449                   1 30 41
     +9    1449 x 9  =   -1 30 41
   ----                   -------
   1458                         0
Figure out how to compute the square root of 2 below...
                         1. 4  1  4  2  1  3
                        /-------------------
    1                 \/ 2.00 00 00 00 00 00
   +1        1 x 1       1 .. .. .. .. .. ..
   --                    ----
    24                   1 00 .. .. .. .. ..
   + 4      24 x 4         96 .. .. .. .. ..
   ---                     -----
    281                     4 00 .. .. .. ..
   +  1     281 x 1         2 81 .. .. .. ..
   ----                     -------
    2824                    1 19 00 .. .. ..
   +   4    2824 x 4        1 12 96 .. .. ..
   -----                    ----------
    28282                      6 04 00 .. ..
   +    2   28282 x 2          5 65 64 .. ..
   ------                      ----------
    282841                       38 36 00 ..
   +     1   282841 x 1          28 28 41 ..
   -------                       -----------
    2828423                      10 06 59 00
   +      3   282842 x 3          8 48 52 69
   --------                       ----------
                                  1 58 06 31

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