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method of finding the square root by long division without using calculator
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method of finding the square root by long division without using calculator

for example to find the square root of 327

first group as 3, 27 . 00, 00, . . .

Start from the leftmost group ' 3 ' .Find the largest non negative integer whose square is just  less than or equal to this 3. here ' 1' is such a number since 12 < 3.

square root of a number by long division without
              using a calculator

carry the next two digits  ' 27 ' down.   
method of finding the square root by long
                  division without using calculator
now double ( multiply by 2 ) the number on top ' 1' and write it at the side  
method of finding the square root by long
                  division without using calculator                                               

now guess / find the greatest positive integer  ' ? '  so that  the product (2?) * (?) is just less than or equal to 227 ( like you do for long division )
for example
 20*0 =0
21*1 =21
22*2 = 44
23*3 = 69
24*4 =96
25*5=125
26*6=156
27*7=189
28*8=224 < 227
29*9=261 > 227

so the digit we require
in place of ' ? ' is 8
write it at the top along with the one to get 18
write 28 * 8 = 224 below the 227 and subtract to get 3

method of finding the square root by long
                      division without using calculator
carry down the next two  digits ' 00'
method of finding the square root by long
                      division without using calculator

note that the decimal point has been crossed

as before double the  18 from the top and write 18*2 = 36 on the side
method of finding the square root by long
                    division without using calculator

as before try to guess the greatest digit '?'  such that (36?) * (?) is less than or equal to 300

360*0 = 0 < 300
361*1 = 361 > 300

so ? = 0
put this digit 0 at the top after 18 remember to put the decimal point also
method of finding the square root by
                          long division without using calculator

now carry the next group ' 00 ' down to get 300 00

as before double the 180 to get 360 (ignore the decimal point) and write 360 at the side

method of finding the square root by long
                    division without using calculator

as before try to guess  greatest digit '?' so that  (360?) * (?) is less than or equal to 30000
3600*0 = 0
3601*1 = 3601 etc
. . .
. . .
. . .
3607*7 =25249
3608*8 =28864 <
30000
3609*9 =32481 > 30000

thereforre ? = 8

put 8  at the top to get 18.08,

put 3608*8 =28864 below 30000 and subtract to get 1136
method of finding the square root by long
                    division without using calculator

now bring down the next group '00' to get 113600

double 1808 to get 3616 to write on the side
method of finding the square root by long
                    division without using calculator

try to guess greatest value of  ' ? ' so that (3616?)*(?) is less than or equal to 113600

36160*0 =0
36161*1 =36161
36162*2 =72324
36163*3 =108489 <
113600
36164*4 =144656 > 113600

therefore ? = 3
square root without calculator

 continue the process till you have the  accuracy you require

here square root of 327 is approximately 18.08


Note  the grouping before you start [ in pairs/(of two) to the left and right ]

3567.23522 will be grouped as  35,67 . 23,52,20

567.23522  will be grouped as  5,67 . 23,52,20

2.3 will be grouped as 2.30,00,00, ...

and so on





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