The Mathematical Miscellany Volume 1-6

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. 1836 Excerpt: ...when the number of signs--is odd. Theorem 3. The product of any number of signs will be the same, in whatever order they are multiplied. An immediate consequence of the preceding definitions, is that the multiplication of signs has no relation to the multiplication of numbers. But we ought not to be surprised that the notion of the product of two signs presents itself as the first step in analysis, since in the addition or subtraction of a monomial, we actually multiply the sign of this monomial by + or by--. The principles thus established will easily enable us to surmount all the difficulties that the use of the signs-and--can present in the operations of algebra and trigonometry. It is only necessary to distinguish carefully the operations relative to numbers from those which refer to quantities, positive or negative. We ought especially to fix in a precise manner the object of both, to define the results produced by them, and to show their principle properties. We shall now endeavour to do this, in few words, for the different operations that are ordinarily used. 19. addition And Subtraction. Sums and Differences of Numbers. The adding of a number A to a number B, or in other words, subjecting the number A to an increment + B, is what we call arithmetical addition. The result of this operation is called sum. It is indicated by placing after the number A its increment B, thus: A + B. We cannot demonstrate, but we may admit as self-evident, that the sum of several numbers is the same in whatever order they are added. This is a fundamental axiom, on which arithmetic, algebra, and all the sciences of calculation rest. Arithmetical subtraction is the inverse of an addition. It consists in taking from a first number A, or second number B, that is, in seeking ...