Let a number be kpqrstm. Observe that square of a single digit number can have up to two digits, square of a two digit number can have up to four digits, square of a three digit number can have up to six digits and square of a four digit number can have up to eight digits. You may have already got a hint now that why we take the numbers in pairs.
As the number has seven digits, so square root will have four digits. And making them in pairs we get ulk" "ul(pq)" "ul(rs)" "ul(tm) and ask is single digit, square root could start from 3,2 or 1.
The numerical value of number is
kxx1000000+pxx100000+qxx10000+rxx1000+sxx100+txx10+m
we an also write it the following way, which we say (A)
kxx1000000+(10p+q)xx10000+(10r+s)xx100+(10t+m)
Let us consider a two digit number abc and let its square root be fg. Actually numerical value of these numbers is 100a+10b+c and 10f+g and hence we must have
100a+10b+c=(10f+g)^2=100f^2+20fg+g^2
or 100a+10b+c=100f^2+ul(2(10f+g))g
Hence, in division method we first search for some f, whose square is equal or just less than a. Naturally f comes in the place for quotient and remainder would be (a-f^2), with place value 100(a-f^2).
For next digit, we choose divisor as double of f (note that its place value is 10f and choose a g, which makes it 10f+g.
I hope this makes this clear. Would have gone for a bigger number like kpqrstm, but things get too complicated.
