Computing

How the manual calculation of a square root works:

Take the solution algebraically as (a + b) (a + b) => a2 + 2ab + b2 => a2 + b(2a +b)

The method pairs the digits from the decimal point and starting with the most significant finds the single integer square root for that pair. The second and continuing operation adds the remainder of that to the next pair. It may be that the most significant pair is only a single digit as in one hundred and forty four 1 44

Lay out the solution as though working on long division with the object under a line and the subsequent divisors to the left below each other.

Example: Find the square root of 40.96

1. Separate the number into pairs of digits from the decimal point, which is simple in this case; rooting a number effectively halves the digits. (40 . 96)
   
   
2. Find the highest number that multiplied by itself will divide into the first 'pair' (40)
   6 x 6=36 (7 x 7=49) so we use 6 - this is 'a'
   
   
3. Place this number, as a part solution above its pair
     6
   40 . 96
   
   
4. Subtract 36 (a²) from the 40 and incorporate the next pair => 4.96
   
   

Now we use the b(2a + b) to resolve b, and can ignore the decimal point in the following processes as each solution is an integer to be placed above it's respective pair.

Now resolve the b(2a +b) from the 496

1. Double a => 12 and move one point left => 12*
2. Find a number such that 12* x * is equal to or less than 496 which in this case is 4 or b
   i.e. 124 x 4 = 496
3. Place this digit above its pair
     6 . 4
   40 . 96

With no remainder the solution is complete 6.4 is the square root of 40.96.

Should there be a remainder continue the process to find another b from the remainder and the next two digits until you find a rational solution or you have a suitably significant number.