CHAP. VI. Restate ment of the indices. FURTHER DEVELOPEMENT OF THE THEORY OF INDICES. 176. OUR first assumption of indices was made for principle of the purpose of abbreviating the expression of the continued product of a symbol (whether arithmetical or not) into itself, the index denoting the number of times that symbol appeared as a factor: thus a2 was taken to denote aa, a3 to denote aaa, and a" to denote the product of a into itself, when repeated a number of times equal to n: under these circumstances, our interpretation of the meaning of the expression a" was confined to those values of n, which were positive whole numbers: and it was deduced as a necessary consequence of this notation, that whenever n and m denoted whole numbers, a" × a" was identical with an+m. (Art. 11, 12.) Taking this conclusion as a guide for our assumptions, we next assumed the existence of expressions such as a", where n was perfectly arbitrary, like all other symbols in Algebra: but in order that the interpretation of the meaning of such expressions should not be arbitrary likewise, when particular values of the index were assigned, we assumed as a general principle that in all cases the product of a" and a", whatever n and m might be, should be equivalent to a"+". It was in conformity with this principle that we shewed that as was equivalent to Va, a equivalent to Va, a equivalent to a or to the cube root of a2, and am equivalent to a", or to the mth root of a", where m and n are whole numbers: by this means a consistent meaning and interpretation was given to all indices, which were positive fractions. (Art. 13.) It is our present object to examine the other consequences of the same principle. indices: a-n 177. Since the index of a" is perfectly arbitrary, Negative it may be negative as well as positive: it remains to equivalent ascertain the algebraical meaning of such expressions as to The general principle of indices, gives a" × a ̃”=a"-", since n m is the algebraical sum of n and -m. Again, if we multiply aa-” into a”, the result by the same prin an am ciple is a"-"+"=a": and if we multiply into a", the result is likewise a", since multiplication and division are inverse operations: it follows, therefore, that a"-" is equi valent to since they both produce the same result when am : multiplied into the same quantity and consequently, it is indifferent whether we multiply a" or any other quantity by a- or divide it by a", the two results being equivalent to each other. m into a is likewise equivalent to a divided by am; it 1 follows therefore that a-m and are equivalent to each other. am an may be 178. This is a most important conclusion, and is Quantities altogether independent of the particular value of the index: transferred it shews that any quantity may be transferred from the from the numerator to the denominator of a fraction by merely to the denochanging the sign of its index and conversely. numerator minator of a fraction, or con versely by changing the signs of their in dices. Proof that is in all cases equivalent to unity, being the representative of a quantity divided by itself: this result is a necessary consequence of the notation. 180. The continued product of am into itself repeated n times, would be denoted by (am)", according to the general principle of indices: and it may likewise be shewn by the same principle, that the same product may be denoted by am", or by a with an index equal to n times the index of the simple factor a": for a x am = am+m =a2m: am×am × am = a3m; and if a" appear four times in the product, the result is a", if five times, it is am, and consequently, if it appears n times, (where n is a whole number) it is denoted by am. We have thus obtained an equivalent form for (am)n, where the symbols m and n are, one or both of them, particular in their representation, though general in their form; it follows therefore, from the law of the permanence of equivalent forms (Art. 131. 132), that am" is equally equivalent to (am)", where m and n are any quantities whatsoever.* * It may be useful to restate, with reference to this case, the reasoning made use of in the establishment of that law: if there be an equivalent form for (am)", it must necessarily be amn, inasmuch as it must coincide with the result obtained in arithmetical Algebra, in conformity with the principle of indices, 181. We subjoin a series of examples which will Examples. explain more fully the consequences of these general conclusions, and will serve to shew the great variety of equivalent forms which we are consequently enabled to give to algebraical expressions. (5) 3a-2x-4a-1x-7a3=84a0=84, since a°=1. indices, when m and n are whole numbers; otherwise the form in question is not a general equivalent form for all values of m and n: and secondly, we may safely assume such an equivalent form of (a")", inasmuch as it coincides with the general form in the subordinate science, with whose rules and operations those of Algebra have been assumed to coincide, when the symbols coincide in signification; and it can never therefore lead to results inconsistent with any other results, which those rules and operations may give. |