Here is an almost-forgotten art, one that, with the advent of electronic calculators, will likely survive to the twenty-first century only on paper and in the memories of oldsters.

What is the number you want to find the square root of? Here's one we'll use:
46656

First, divide the number to be square-rooted into pairs of digits, starting at the decimal point. That is, no digit pair should straddle a decimal point. (For example, split 1225 into "12 25" rather than "1 22 5"; 6.5536 into "6. 55 36" rather than"6.5 53 6".)

Then you can put some lines over each digit pair, and a bar to the left, somewhat as in long division.

+--- ---- ----
| 4 66 56

Find the largest number whose square is less than or equal to the leading digit pair. In this case, the leading digit pair is 4; the largest number whose square is less than or equal to 4 is 2.

Put that number on the left side, *and* above the first digit pair.
2
+--- ---- ----
2 | 4 66 56

Now square that number, and subtract from the leading digit pair.

2
+--- ---- ----
2 | 4 66 56
|-4
+----

0 Extend the left bracket; multiply the last (and only) digit of the left-hand number by 2, put it to the left of the difference you just calculated, and leave a decimal place next to it.

2
+--- ---- ----
2 | 4 66 56
|-4
+----
4_ | 0

Then bring down the next digit pair and put it to the right of the difference.

2
+--- ---- ----
2 | 4 66 56
|-4
+----
4_ | 0 66

Find the largest number to put in this blank decimal place such that that number, times the number already there plus the decimal place, will be less than the current difference. For example, see if 1 * 41 is <= 66, then 2*42, etc. In this case it's a 1. Put this number in the blank you left, and in the next decimal place on the result row on the top.

2 1
+--- ---- ----
2 | 4 66 56
|-4
+----
41 | 0 66

Now subtract the product you just found.

2 1
+--- ---- ----
2 | 4 66 56
|-4
+----
41 | 0 66
|- 41
+--------
25

Now, repeat as before: Double the last digit on the left; bring down the next digit pair on the right; find the largest next result digit; subtract.

2 1
+--- ---- ----
2 | 4 66 56
|-4
+----
41 | 0 66
|- 41
+--------
42_ 25 56

(426 * 6 = 2556)
2 1 6
+--- ---- ----
2 | 4 66 56
|-4
+----
41 | 0 66
|- 41
+--------
426 | 25 56
|- 25 56
+-------------
0

When the difference is zero, you have an exact square root and you're done. Otherwise, you can keep finding more decimal places for as long as you want. Blogdup