ROOT. In the ancient arithmetike (Icpc0Arruci7),—the theory of numbers of the Greeks—the numbers considered were primarily integers. A square number was therefore an integer that had two equal integral factors; for example, 9 was a square number, the product of the two equal factors 3 and 3. These factors repre sented the length of the side of a square of which the area is 9 square units. Each was therefore called a "side," and so the Latins spoke of "finding" a latus as modern writers speak of "extracting" a square root. The Arab writers of the 9th century spoke of such a factor as root, and their mediaeval translators adapted the word radix (compare "radish") with the adjective radical and the verb extrahere (pull out). In modern usage a root is one of the equal factors of any kind of a number or of an algebraic expression, the square root being one of the two equal factors : the cube root, one of three ; the fourth root, one of four, etc. Thus the cube root of 216, 0.343 and — are respectively 6, I-, 0.7 and It is also customary to use such expressions as "the square root of 2," although 2 has no two equal factors, meaning that this root is approximately 1.414 . . . and designating the number of decimal places (or of significant figures) to which it is to be carried. Symbolically, V2 or 2 means the square root of 2, and 2 or 21 means the cube root of 2, but see the "principal root" referred to later. The index of the root (indicating what root is to be found) is not written in the square-root symbol but it is inserted in the case of other roots. The expression "root 2" means in Great Britain the square root of 2, but in the United States the word "square" is used be fore "root" in this connection.
In the modern extension of the number system, every number has two square roots, one positive and the other negative; thus the square root of 4 is +2 or — 2. It also has three cube roots;
thus the cube root of i is 1, —3, and —1.—h/ as can be seen by cubing these numbers. Similarly, the number of nth roots of a number is n, of which (if n is even) two are real and the rest are imaginary (or complex; see Complex Numbers) ; or of which (if n is odd) one is real and the rest are imaginary. For example, the four fourth roots of 16 are +2, —2, +2V —I, and —2\/ — I ; or as ordinarily written, LT__.2i; and the three cube roots of 8 are 2, —3.
The symbol V means the principal root of the number. Thus, V4 means 2, not — 2; and 8 means 2, not — The principal root of a positive number means the positive real root.
The common method of finding any root of a number is based upon the Binomial Theorem (q.v.) and is given in most text books on arithmetic and elementary algebra. Practically, how ever, in working physical problems, square and cube roots are found from tables (q.v.), by logarithms (q.v.) or by the use of such instruments as the slide rule or other computing machines. (See MATHEMATICAL INSTRUMENTS.) A root of an equation is a value of the unknown quantity that will reduce the equation to an identity. For example, 3 is a root of the equation 21 = 0 ; for if 3 be substituted for x, we have 9+12-21=-0. There are two roots to this equation, the other being — 7. Every rational integral algebraic equation has as many roots as the degree of the equation; for example, an equation of the fifth degree has five roots, not necessarily different and not all necessarily real. If unreal, the imaginaries (complex) roots enter in pairs. (See EQUATIONS, THEORY OF.)