IRRATIONAL

In Arithmetic and Algebra, a term applied to numbers or quantities of which the roots are incommensurable with unity, and therefore cannot be accurately extracted. Thus, the root y^ is irrational, because it cannot be expre ed by any finite number. If the side of a square = 1, then V^ is its diagonal; and it is proved by elementary geometry that the diagonal of a square is incommensurable with its side. Irrational quantities are also called surds. IRREDUCIBLE CASE, in Algebra, is that particular case in the solution of a cubic equation in which Cardan's celebrated formula contains an imaginary expre ion, and therefore fails in its application. This circumstance occasioned great embarra ment to the early analysts, and all the efforts of their succe ors to overcome the difficulty in a direct way have proved unsucce ful. In order to show in what it consists, let the proposed cubic equation bej:3 + aa: + c = 0; then, by Cardan's rule, we have x = (-^ c + V5V«^ + ic")^+(~ ^''~ VaV^'+i^')** Now, if in this exp'e ion a is negative, and ^^ a^ is greater than i c^; then ^■j c^ + \ c'^ will be a negative quantity, and consequently the extraction of its square root will be impo ible, or the expre ion y i ^3.|. i ^s will be imaginary. But it is known from the theory of equations that every cubic equation must have at least one real root; and it is a circumstance not a little remarkable that those cubic equations in which this imaginary expre ion occurs have not only one real root, but have all the three roots real. It is po ible to disengage the expre ion for the value of;r from the imaginary quantities by expanding it by the binomial theorem; for the imaginary quantities, which will be the same in both the resulting series, will be positive in the one series and negative in the other; and therefore, on adding the series together, they will be eliminated. But the series which results from this addition will very rarely be convergent, consequently this method is of no use whatever. Various indirect methods of finding the roots have been proposed; the following is one of the simplest: — Let y^ — q 1/ = r he the proposed equation. Find in the trigonometrical tables an arc a whose natural cosine is 3 r V^ -T- 2 5 V? > then the three roots of the proposed equation are, y = 2^1 qvi^coa. is, y = - 2 Vi? X sin. i (90° - a), jf = — 2 Vs? X sin. i (90O + a). These formula apply, whether r be positive or negative; but when r is negative, it is more convenient to choose the arc a such that its natural sine is 3 r \/3 -i-2q \/q, and the three roots are_then y = 2 \/^ g X sin. | a, y = 2 VI? X cos. i (90O -|- «), y = - 2 V§? X cos. A (90O - d) See the article " Algebra," by Profe or Wallace, in the Ency. Britannica. [s. 626]