Square Roots By Hand

This is how it was done before pocket calculators

If you need a square root, the easiest and most accurate way is to use a calculator. Before calculators, other methods had to be used. Logarithms make fast and easy work: find the log, divide by 2, and find the antilog. If you needed more accuracy, or tables were not available, an arithmetic method, or algorism, was available. This was taught in American schools until around the Second World War, then disappeared as being too hard, even before calculators. The method isn't hard at all, and might be of some curiosity value, so here it is.

Involution, or extracting a square root, looked like long division, which, in fact, it was, except that the divisors changed as the solution progressed. It is much easier to explain by example than by theory. Suppose we require the square root of 258.2449. The calculation is shown at the left. The number is written down, and groups of two digits are marked off to right and left of the decimal point, ending with either one or two digits. Each group corresponds to one digit in the square root. We first determine the largest integer whose square is less than the value of the first group, in this case 1, which we write to the right (some people write it above), where the square root is accumulated, and also to the left, where the divisors are kept. It is squared and written under the first group, from which it is subtracted, in this case with remainder 1. The next group is brought down to form the number 158. The 1 is doubled, obtaining 2, and a zero is added. This is the trial divisor. It is divided into 158, and the quotient is 7. Now 7 is added to the trial divisor, and the sum is multiplied by 7 and subtracted from the 158. But 27 x 7 = 189 -- too big! Well, try 6 instead of 7. Now, 26 x 6 = 156, which is less than 158 and therefore all right. The actual divisor, 26, is written down, and 6 is added to it again to get 32. A zero is added to get the trial divisor 320, and the next group is brought down to form the dividend 224. 0 must be the next digit, we see, since no larger number will do, so we add it to the square root. The next group, 49, is brought down, and a zero is appended to the 320 to get the trial divisor 3200. This goes into the dividend 7 times, and, in fact, 3207 x 7 = 22449 exactly, so the square root is 16.07 exactly.

The process for most square roots does not terminate, and the procedure is continued until the desired accuracy is obtained. Finding the square root of 3 is such a problem, and its solution out to 8 places is shown. A four-function calculator can be used to help carry out this procedure, by the way. There is a similar method for cube roots, but it is a little trickier. There are more concise ways of writing down the problem, but this layout makes the construction of the divisors clearer. The answer is checked by squaring it, of course. Algorism always requires a check -- it is too easy to make mistakes.

Composed by J. B. Calvert 1999
Last revised 18 December 2004