What are said to be homogeneous. (a + b) (c + d), means that the whole quantity a + b is to be incorporated by multiplication into c + d. The connection indicated by the vinculum, when not very distinctly written, is in some cases ambiguous, which is never the case with brackets, which are therefore generally adopted. 19. Quantities are said to have the same dimensions when quantities the sum of the indices of the symbols incorporated in each of their terms is the same, unity being supposed to be the index of each simple symbol such as a, b, c, &c.: such quantities are likewise said to be homogeneous: thus a is homogeneous with b, c, x, x: a b is homogeneous with a2: abc is homogeneous with a2b and a3: a2x-ax2 is homogeneous with a3-3 x y z + ≈3, and similarly in other cases. 20. We likewise speak of quantities as being of one, or n dimen. two, three or n dimensions, the dimension being measured sions. by the sum of the indices of the symbols involved: thus a is a quantity of one dimension: xy and a2 are quantities of two dimensions: xyz and a3 are quantities of three dimensions, and æ”, în-1y, xn−2 y2, &c. are quantities of n dimensions. Quantities of one, two, Not affected cal coeffi. 1 A numerical coefficient does not affect the dimensions by numeri- of a quantity, since it alters its magnitude only, and not its cients. nature the same remark may be made with respect to any symbol, which is assumed to represent an abstract number. like or un like. Quantities 21. We speak of algebraical quantities as like or unlike, according as they involve the same or different symbols, without regarding their signs or numerical coefficients: thus 2 a and -3a, -2 and 4x2, 5abc and -7abc, are pairs of like quantities: a, b, c, x2, 4ab, a3, 7abc, are unlike quantities. Meaning of 22. The sign, placed between two quantities or the sign expressions, indicates that they are equal or equivalent to each other it may indicate the identity or absolute equality of the quantities between which it is placed or it may shew that one quantity is equivalent to the other, that is, if they are both of them employed in the same algebraic operation, they will produce the same result: or it may simply mean, as is not uncommonly the case, that one quantity is the result of an operation, which in the other is indicated and not performed. 23. In order to indicate that one quantity is greater or less than another, we write them consecutively with the sign > between them in one case, and < in the other: thus a>b means that a is greater than b: and a <b means that a is less than b. and terms. 24. There are other signs and terms, which we shall Other signs have occasion to use, the explanation of which may be most conveniently reserved for those parts of the subject where they first occur. Amongst the various definitions, assumptions and propositions contained in the preceding Articles, there are many which are imperfectly stated, and which would require some acquaintance with the practice of Algebra, to make their complete developement and accurate limitation intelligible: it is on this account that the further discussion of them has been reserved for a more advanced part of the subject. B CHAP. II. Addition. What is meant by the sum of ON THE METHODS OF COMBINING AND INCORPORATING ALGEBRAICAL QUANTITIES BY THE OPERATIONS OF AND 25. THE operation of Addition is denoted by the sign +, which, when combined with the signs of each of the quantities to be added, leaves them the same as before (Art. 7.). RULE. Algebraical quantities, therefore, whether simple or compound, are added together by simply connecting them with their proper signs. Like algebraical quantities (Art. 21.), must be collected into one term, whose coefficient will be the difference with its proper sign of the sums of the coefficients of the positive and negative terms respectively. In performing this operation, the quantities to be added are either written in one continued line, or placed underneath each other, as in the addition of numbers in arithmetic the like quantities are then collected severally into one term, and the whole result written in one line. 26. By the sum of two quantities in Algebra, we mean the result of their addition to each other, according to the two quan- preceding rule: thus the sum of a and b is a+b: of a and b, is a + (- b) : = a b: of tities in Algebra. a and −a + (-b) = −a-b: and similarly in other cases. b, is The first of these examples might be written thus, a+a2a: or preserving all the signs, +a++a: the first is omitted as being understood, without being written the two signs ++, which come together, are replaced by +, according to the rule (Art. 7.) These might be written a - a and a + a, respectively: and inasmuch as addition and subtraction are inverse operations (Art. 10.), the results are severally equal to In the two first of these examples, the coefficients of the like quantities 3a and 5 a have the same sign, and their arithmetical sum must, therefore, be taken with its proper sign in the two last, the signs of 3 a and 5 a are different, and the arithmetical difference of their coefficients must be taken with the sign of the greater. In these three examples, the quantities to be added are severally like quantities (Art. 21.): the first example is equivalent to 3a + 7 a -5a4a, and, therefore, to 10 a 9 a or a, the sum of the positive coefficients being 10, and of the negative coefficients being 9: in a similar manner, the second example is equivalent to 3x2 – 12 x2, or 92 and the third example to 25 abc - 21 abc, or 4 abc. : The symbol b disappears, being both added and subtracted (Ex. 3 and 4.); the result in this case, when expressed in words, gives the following general proposition: "If to the sum of any two quantities we add their difference, the result is equal to twice the greater." The quantity in the first of these examples, and ab in the second, disappear in the results. |