long hand square root

Aryabhata

Aryabhata example (decimal)
       ---------
    \/ 8 0 1 2 3  =  ?  (say to 2 decimal places)


        2   8   3 . 0   6     <- x
       -------------------
    \/ 0 8 0 1 2 3.0 0 0 0    <- n
2        4
       ---
         4 0 1
48       3 8 4
         -----
           1 7 2 3
563        1 6 8 9
           -------
               3 4 0 0
5660
               -------
               3 4 0 0 0 0
56606          3 3 9 6 3 6
               -----------
                     3 6 4   <- r

... so n = x*x + 0.0364
Aryabhata example (binary)
        ------------------
      \/ 1 1 0 1 0 1 1 0 1   =  ?


        1   0   1   0   0      <- x
      --------------------
    \/ 0 1 1 0 1 0 1 1 0 1     <- n
1        1
       ---
         0 1 0
100
         -----
           1 0 1 0
1001       1 0 0 1
           -------
                 1 1 1
10100
                 -----
                 1 1 1 0 1
101000
                 ---------
                 1 1 1 0 1    <- r

... so n = x*x + r