First point off the given number into periods of two figures each, beginning with the decimal point and proceeding to the left and right. The following numbers are thus pointed off: 12703, 1'27'03; 12.703, 12.70'30; 220000, 22'00'00; .000442, .00'04'42.

* A cipher is not a digit.

Having pointed off the number, move the decimal point so that it will fall between the first and second periods of the significant part of the number. In the above numbers, the decimal point will be placed thus: 1.2703, 12.703, 22, 4.42.

If the number has but three (or less) significant figures, find the significant part of the number in the column headed n; the square root will be found in the column headed or according to whether the part to the left of the decimal point contains one figure or two figures. Thus, = 2.1024, and = 4.6904. The decimal point is located in all cases by reference to the original number after pointing off into periods.

There will be as many figures in the root preceding the decimal point as there are periods preceding the decimal point in the given number; if the number is entirely decimal, the root is entirely decimal, and there will be as many ciphers following the decimal point in the root as there are cipher periods following the decimal point in the given number.

Applying this rule, = 469.04 and =

.021024.

The operation when the given number has more than three significant figures is best explained by an example.

(a) =? (b) =?

### Solution

(a) Since the first period contains but one figure, there is no need of moving the decimal point. Look in the column headed n2 and find two consecutive numbers, one a little greater and the other a little less than the giver number; in the present case, 3.1684 = 1.782 and 3.1329 = 1.772. The first three figures of the root are therefore 177. Find the difference between the two numbers between which the given number falls, and the difference between the smaller number and the given number; divide the second difference by the first difference, carrying the quotient to three decimal places and increasing the second figure by 1 if the third is 5 or a greater digit. The two figures of the quotient thus determined will be the fourth and fifth figures of the root. In the present example, dropping decimal points in the remainders, 3.1684 - 3.1329 = 355, the first difference;

3.1416 - 3 1329 = 87, the second difference; 87/355 = .245+, or .25. Hence, = 1.7725.

(6) =? Pointing off into periods we get 23'42.90; moving the decimal point we get 23 4290; the first three figures of the root are 484; the first difference is 23.5225 - 23.4256 = 969; the second difference is 23.4290 - 23.4256 = 34; 34/969 = .035+, or .04. Hence, = 48.404.