Square Root (2004 html file)Updated: December 11th, 2020

Created: 10/12/20

Manual Calculation of Square Roots

n^
0.5 (n is a real number)

You can, and hopefully will, try this process using any number and then check the result
otherwise. The 'reasoning' behind this method is available - Ref: END NOTES.

First the number to be 'rooted' is formatted into
digit pairs either side of the decimal point, adding pairs of
zeros if further accuracy is required. For example : A solution
to three decimal point requires four pairs of digitits after the
decimal point.

To find the square root of 623.45
to 2dps we would format the 'subject number' as
follows in row 1. of the following table, then process as per the
remaining rows.

pan="7">
>
Row ses Result Use

Method

1. Format Subject Number (623.45)
and place the decimal point.		 6'23.45'00'00 Place formated number
as in long division		         .
  6'23.45'00'00
2. Find highest integer that when squared

is <= the first 'pair' of digits.		 2 Place result above
it's digit pair		   2    .
2 6'23.45'00'00
3. Multiply the integer above by the result 2 x2 = 4 Subtract the result from
the first 'pair' of digits.		   2    .
2 6'23.45'00'00
4
2
4. Bring down the next pair of digits. 23 Thie new subject is 223   2    .
2 6'23.45'00'00
4
2'23
5. Multiply the existing result (2) by 2 2 x 2 = 4 Place it to the left with
space for a second digit		   2    .
2

4_		 6'23.45'00'00
4
2'23
6. Find the largest 'a' where
((fourty+a) x a) <= the subject		 a=4

44 x 4
= 172		 Place the '4' above the
second pair of digits		   2  4.
2

44		 6'23.45'00'00
4
2'23
7. Multiply the integer above by the result 176 Subtract the result from
the new subject (223)		   2  4.
2

44		 6'23.45'00'00
4
2'23
1'76
   47
8. Bring down the next pair of digits. 45     2  4.
2

44		 6'23.45'00'00
4
2'23
1'76

   47'45
9. Multiply the existing result (24) by 2 48 Place it to the left withspace
for another digit		   2  4.

2

44

48_

 6'23.45'00'00
4
2'23
1'76

   47'45
10. Find the largest 'a' where
((fourhundred and eighty+a) x a)
is <= the subject		 a=9

489 x 9
=4401		 Place the 'a' above the
next pair of digits		   2  4. 9
2

44

48_		 6'23.45'00'00
4
2'23
1'76
   47'45
11. Multiply the integer 'a' by the result 4401 Subtract the result from
the new subject (4745)		   2  4. 9

2

44

489

 6'23.45'00'00
4
2'23
1'76

   47'45
   44'01
     3'44
12. Bring down the next 2 digits. 00 Subtract the result from
the new subject (4745)		   2  4. 9

2

44

489

 6'23.45'00'00
4
2'23
1'76

   47'45
   44'01
     3'44'00
13. Proceed as per 9 onwards.
Then 10. Double the 'answer so far'
Find the largest 'a' where
((four-thousand nine-hundred and eighty+a) x a)
<= the new subject (34400)		 2 x 249
= 498		     2  4. 9
2

44

489

498_		 6'23.45'00'00
4
2'23
1'76

   47'45
   44'01
     3'44'00
END NOTES :- Ok that's the general idea.

Although I have known this process for over 40 years, it is only in the last few weeks (2004) that I discovered how the above process worked during my latest attempt to discover a method to find the third root of a number.

All the best - Roger Lovejoy - 7th November, 2004

 
print   Print