Choose any number (y) smaller than the number (m) for which you must calculate the principal square root. For example, if required to find the principal square root of m = 9, choose any number (y) that it is smaller than 9. A suitable choice for y would be y = 4.
Calculate the average (v) of y and (m/y). The average (v) = [y + (m/y)]/2. For example, if y = 4 and m = 9, then v = [4 + (9/4)]/2 = 3.125.
Replace the value of y with the value for v calculated in step 2. For example, if v = 3.125, then the new value for y = 3.125.
Repeat steps 2 and 3, until the values for (y) and (m/y) are approximately equal. For example, if y = 3.125, then (m/y) = (9/3.125) = 2.88. The difference between (y) and (m/y) = 3.125-2.88 = 0.245. Because this difference is greater than zero, steps 2 and 3 are repeated.
Report the principal square root of the number m as the final value for y when y is approximately equal to (m/y). The principal square root is always a positive number. For example, when y = 3, then (m/y) = (9/3) = 3. The principal square root of m = 9 is +3.