calculate the Square Root using Pencil and Paper

Introduction
In this article the ancient art of calculating a square root, using pencil and paper, is rediscovered.
The reader should know the addition and multiplication tables by heart .

After the examples an explanation follows why this method is correct.

Notation: the square root of 100 is written as SQRT(100).

Examples
1.
We calculate SQRT(4096)

step 1: step 2: step 3: 
	SQRT(40 96)  =  6
	     36
	     ---
	      4 96
The temparary answer is 6. The remainder is 496

Step 4: Step 5: step 6: step 7: 
	W(40 96)  =  64
	  36
	  ---
	   4 96
	   4 96
	   ------
	      0
SQRT(4086) = 64, because 64 *64 = 4096

2.
Calculating SQRT(1522756)

- split number in groups of 2 digits, add 0 when number of digits is odd - find first square = 1 
	SQRT(01 52 27 56)  =  1
	      1
	     ---
	      0 52
- multiply by 2 : 2 * 1 = 2
- write 2? * ?
- ? = 2
- 22 * 2 = 44
	SQRT(01 52 27 56)  =  12
	      1
	     ---
	      0 52
	        44
	        ---
		   8 27
- multiply temporary answer by 2 : 2 * 12 = 24
- write 24? * ?
- ? = 3, because 243 * 3 = 729 
	SQRT(01 52 27 56)  =  123
	      1
	     ---
	      0 52
	        44
	        ---
		   8 27
		   7 29
		   ------
		     98 56
- multiply temporary answer by 2 : 2 * 123 = 246
- write 246? * ?
- find ? = 4 ( 3 too small, 5 too large)
- 2464 * 4 = 9856 
	SQRT(01 52 27 56)  =  1234
	      1
	     ---
	      0 52
	        44
	        ---
		   8 27
		   7 29
		   ------
		     98 56
		     98 56
		     -----
		         0

3.
Calculate SQRT(5).
Because 5 is no square, the answer will be an approximation.
- write 05| . 00 | 00 | 00 ................
- first square = 2 
	SQRT(05)  =  2.
	      4
	     ---
	      1 00
place decimal point in answer, but do not use in calculations.

- double answer : 2 * 2 = 4
- write 4? * ?
- ? = 2
- 42 * 2 = 84
	SQRT(05)  =  2.2
	      4
	     ---
	      1 00
	        84
	        ---
		   16 00
- double answer : 2 * 22 = 44
- write 44? * ?
- find ? = 3
- 443 * 3 = 1329
- subtract
	SQRT(05)  =  2.23
	      4
	     ---
	      1 00
	        84
	        ---
		   16 00
		   13 29
		   -----
		    2 71 00
- double answer : 2 * 223 = 446
- write as 446? * ?
- ? = 6
- 4466 * 6 = 26796
	SQRT(05)  =  2.236
	      4
	     ---
	      1 00
	        84
	        ---
		   16 00
		   13 29
		   -----
		    2 71 00
		    2 67 96
		    -------
		       3 04
- double answer : 2 * 2236 = 4472
- write as 4472? * ?
- ? = 0, so only shift down 2 zero digits
	SQRT(05)  =  2.2360
	      4
	     ---
	      1 00
	        84
	        ---
		   16 00
		   13 29
		   -----
		    2 71 00
		    2 67 96
		    -------
		       3 04 00
- reapeat steps before for more digits in answer.

Why this works
Core of the method is the product For clarity this notation is introduced:
if a number consists of digits a and b then so Example : SQRT(4096) again.
Let the answer be [ab] = 10a + b so We have to find digits a and b.
Write number as 40.96 but this is exactly what we did before, finding the value of ? in 12? * ? .
Only, b is used instead of ?

Another example
SQRT(01 52 27 56)
Write as SQRT( 01 . 52 27 56) and we observe The temporary answer [12].
Write original number as 152 27. 56
Multiply temporary answer by 10, result is [12b]
So far, number 1 52 27 . 56 - [120]2 is the remainder: And this is the tric:
regard [12] as the new value of a and find value of b in [12b]

Following instructions above, we find b = 3, so the new temporary answer is [123]
Call [123] the new value of a.
Again shift decimal point 2 places right and multiply temporary answer by 10
Calculated value b = 4 and remainder is 0.

The general explanation
Say the root of an 8 digit number xx xx xx xx must be calculated. Do not worry about tehe decimal point : it is automatically right.
Two digits of the number produce one digit of the answer.
The purpose of the decimal point was only to facilitate the explanation.