Pure Mathematics Volume 1; Including the Higher Parts of Algebra and Plane Trigonometry, Together with Elementary Spherical Trigonometry (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1877 Excerpt: ...result that it shall agree exactly with the proposition at the head of this article. For we may conceive of a3 as representing the product of unity and the quantity a3. We shall therefore be perfectly consistent if we allow a3 to represent the quotient of unity by the quantity a3. We shall then have a_s =-, and hence we get from the a3 above result--a4 + a7 = a3 = a47. Or, generally, m being less than n, we have--am aaa to m factors 1 a" aaa to n factors aaa (n--m) factors =------; or, using the flotation just explained, a 3. Let the powers of the quantities in the dividend and divisor be equal. It is evident that their quotient is unity. m, a aaaa 1-, Thus, -= = = 1. a aaaa 1 am And so, --= 1. If, however, we assume the principle proved in the two cases above to hold here, we have--nm--= amm = a." am It follows therefore that a" = I. Cor.--From the above interpretation of negative indices it follows that the same rules for multiplication and division of quantities involving them may be applied as in the case of positive indices. Ex. 5. Divide 3 a + 13 ab-10 62 by a + 5 6. When the divisor, as in this example, contains more than one term, it is generally convenient to follow the method of arithmetical long division. Thus--a + 5 6) 3 a2 + 13 a6-10 62(3a-26 3 a2 + 15 ab-2a6-10 62-2a6-10 62 Ex. 6. Divide a + -262 + 64 by a? + ab + 62. a2 + ab + 62) a4 + a262 + 64 (a2-ab + 6 a4 + as6 + a262-a3b + 64-a36-a262-ab3 a262 + a68 + 64 a262 + ab + 64 It will be seen that in the last two examples care has been taken to keep the terms of the divisor, dividend, and successive remainders arranged according to the ascending or descending powers of some letter. In these cases we have arranged the terms according to the descending powers of -, and, as t...