

Calculate square root with 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:
split the number (rigth to left) in groups of 2 digits, result 40  96
step 2:
find the digit which square is closest (below or equal) to 40,
this is the 6 , because 6 * 6 = 36
step 3:
subtract 36 from 40 and shift next 2 digits in place
SQRT(40 96) = 6
36

4 96
The temparary answer is 6. The remainder is 496
Step 4:
multiply temporary answer by 2, so 2 * 6 = 12
Step 5:
step 6:
find digit ? for value closest (below) or equal to remainder.
This digit is 4 , because 124 * 4 = 496
step 7:
subtract and add digit 4:
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
(a + b)^{2} = a^{2} + 2ab + b^{2}
For clarity this notation is introduced:
if a number consists of digits a and b then
so
[abc] = 100a + 10b + c
[23b] = 230 + b
[(2*3)b] = [6b] = 60 + b
Example : SQRT(4096) again.
Let the answer be [ab] = 10a + b so
4096 = [ab]^{2} = (10a + b)^{2} = 100a^{2} + 2*10*ab + b^{2} = 100a^{2} + (20a + b)b
We have to find digits a and b.
Write number as 40.96
6^{2} = 36 < 40 and 7^{2} = 49 > 40,
a = 6
[6b]^{2} = 36 00 + 2*10*6b + b^{2} = 4096
120b +b^{2} = (120 + b)*b = [12b]*b
[12b]*b = 4096  36 00 = 496
b = 4
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
a = 1
write number as 1 52.27 56
[1b]^{2} = 1 00 + 2*10*b + b^{2} = 152 ,27 56
20b + b^{2} = (20 + b)*b = [2b]*b
[2b]*b = 1 52, 27 56  100 = 52, 27 56
b = 2
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:
1 5227, 56  14400 = 827, 56
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
1 52 27 56  12 30^{2} = 9856
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.
step  action 
1  write number as xx . xx xx xx = a . .^{2} 
2  calculate a (temporary answer) and subtract : remainder = number  a^{2} 
3  multiply remainder by 100, temporary answer by 10 
4  calculate b in [ab] 
5  calculate new remainder = remainder  [(2a)b]*b 
6  new a = [ab] 
7  repeat steps 3 .. 7 
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.

