Home >> Encyclopedia Americana, Volume 25 >> Sponge to Statistical Method >> Square Root

# Square Root

## additions, surd, tens and quantity

SQUARE ROOT. The square root of a number is one of its two equal factors. It is indicated by the fractional exponent (1/2) placed at the right, and above the number, thus 16i, or by the radical sign ( The two equal factors of 16 are 4 and' 4, either one of which may be taken as its square root. The square roots of many numbers are approximate only and are repre sented by a whole number and a decimal, the latter carried out as many places as the ap proximation is desired, as example, V 19= 4.358899+. The square root of fractions may be found by extracting the root of the numera tor and denominator, but a more practical method is to extract the root of the resulting decimal. Illustration of the method employed in finding the square root of 576: 576(20 20=400 4 2 X 20=40)176 24 (40 + 4) X4=176 Since the number 576 has three figures its square root will be composed of tens, and units. The number of tens in the root will be 2, and the square of 2 tens, or 20, will be 400. (See Fig. 1). But inasmuch as there is still a re mainder of 176, such additions must be made to the square as will take up this remainder, and still keep the figure a perfect square. The neces sary additions are the two rectangles B and C and the small square D (see Fig. 2). The re mainder 176 divided by the length of the rec tangles, 2 X 20, will give the width of the addi tions, which is 4, and this width is also the side of the small square D; therefore the total length of the additions will be 40 +4, and the area of the additions 4 times this length, or 176, which completes the square whose area is 576, and whose square root is 24.

The preceding rules, with the exception of those relating to decimals, are ap plicable to algebraic quantities. The square root of an algebraic quantity, however, may be positive or negative.

The square root of a negative quantity is imaginary, and is usually factored into two quantities, one of which is real, and the other expressed by V — 1. Thus the square root 1/ ab V —1.

The square root of algebraic quantities af fected by other roots is indicated by multiplying its exponent, or index by 2, wherever possible the square root is first extracted, and the multi plication avoided.

The indicated square root of an imperfect square, in algebra, is called a quadratic surd. To extract the square root of a binomial surd, such there are many methods ; one is to reduce the surd term so that its co efficient shall be 2. Then separate the rational term into two parts whose product shall be the quantity under the radical sign. Extract the square roots of these parts, and connect them by the sign of the surd term.

Square root finds its application in all branches of mathematics, and in the natural sciences. Its use is fundamental.