SURD, in arithmetic and algebra, de notes any number or quantity that is in commensurable to unity ; otherwise call ed an irrational number or quantity.
The square roots of all numbers, except 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, &c. (which are the squares of the integer numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, Etc ) are incommensurable.: and after the same manner the cube roots of all numbers, but of the cubes of 1, 2, 3, 4, 5, 6, &c. are incommensurables: and quantities that are to one another in the proportion of such numbers must also have their square roots, or cube roots, incommensurable.
The roots, therefore, of such numbers, being incommensurable, are expressed by placing the proper radical sign over them : thus v 2, 3, v 5, / 6, &c. express numbers incommensurable with unity. However, though these numbers are incommensurable themselves with unity, yet they are commensurable in power with it ; because their powers are integers, that is, multiples of unity. They may also be commensurable sometimes with one another, as the 8 and t/ 2; because they are to one another as 2 to 1: and when they have a common mea sure, as v 2 is the common measure of both ; then their ratio is reduced to an expression in the least terms, as that of commensurable quantities, by dividing them by their greatest common measure. This common measure is found, main com mensurable quantities ; only the root of the common measure is to be made their common divisor : thus 1/12 = = 2, a and — 3 A rational quantity may be reduced to the form of any given surd, by raising the quantity to the power that is denominat ed by the name of the surd, and then setting the radical sign over it : thus a = Vi a' a3 v =_- v am, and 4 = = ,y64 = 1/256 V1024 = vsla.
As surds may be considered as powers with fractional exponents, they are re duced to others of the same value, that shall have the same radical sign, by re ducing these fractional exponents to frac tions having the same value and a com mon denominator.
1 Thus Iv a .-. and 0, y a =_- a ',7„ and 1 In 1 TI =. •=;-- = and therefore va and a, reduced to the same radical sign, become "'",/ a and 7/ a n. If you are to
reduce V 3 and 4/ 2 to the same deno minator, consider v 3 as equal 31, and 4/ 2 as equal to 21, whose indices, re duced to a common denominator, you have 31 = and 21 21, and con sequently, v 3 = v 33, = and 1y 2= V 2 a= 4; so that the pro posed surds V 3 and 4/ 2, are reduced to other equal surds V 27 and 4, hav ing a common radical sign.
Surds of the same rational quantity are multiplied by adding their exponents, and divided by•subtracting them : thus, ass V a X a al X a- a = a 6 =a 1' o a j--1 ; and — = a a — at 5-3 a 12 = a ..1 V a ; a xtiss m+r, n..! n m a ; =. a ; 2 X 4/ m t/ 2.
VS2 se 2 If the surds are of different rational quantities, as 'V a' and ',/ 63, and have the same sign, multiply these rational quantities into one another, ordivide them by one another, and set the common radi cal sign over their product, or quotient. Thus Z./ a' x = a' ; t/ c0 se.
2 X V 5 = V 10 ; =_ a co = = b3 g V 3.
If surds have not the same radical sign, reduce them to such as shall have the same radical sign, and proceed as before : m/a x b =a "V am Iss; a = ,n Z/ X iza 2x X 411= V2 3 X 4' = ?8 x 16 = t/1 2 8; 17 — 411.— 65.; 21 sl T V2 =-- 6 / 16 = V2.
%I 8 If the surds have any rational coeffi cients, their product or quotient must be prefixed; thus, 2 i/ 3 X 5 V 6= 10 I/ 18. The powers of surds are found as the powers of their quantities, by multi plying their exponents by the index of the power required; thus the square of 2 2 is 21 21 4 ; the cube of 1i3 t/ 5 = 51 V 125 Or you need only, in involving surds, raise the quantity under the radical sign to the power required, continuing the same ra dical sign; unless the index of that power is equal to the name of the surd, or a mul tiple of it, and in that case the power of the surd becomes rational. Evolution is performed by dividing the fraction, which is the exponent of the surd, by the name of the root required. Thus, the square root of .0/(74 is or Z/iTti.