EXAMPLES FOR PRACTICE. 1. Required the sum of (a+b) and (a+b) 2. Add 5x-3a + 6 + 7 and ther. 4a 3. Add 2a + 3b 4c 9 and 5a-3b+2c ther. 4. Add 3a+2b — 5, a + 5b gether 3x+2b-9 toge 5 Add x2+ax2 + bx +2 and x3+cx2+dx-1 together. SUBTRACTION. SUBTRACTION is the taking of one quantity from another; or the method of finding the difference between any two quantities of the same kind; which is performed as follows (b) : RULE. Change all the signs (+ and -) of the lower line, or quantities that are to be subtracted, into the contrary signs, or rather conceive them to be so changed, and then collect the terms together, as in the several cases of addition. (b) The term subtraction, used for this rule, is liable to the same objection as that for addition; the operations to be performed, being frequently of a mixed nature, like those of the former. 1. Find the difference of (a+b) and (ab). MULTIPLICATION. MULTIPLICATION, or the finding of the product o' two or more quantities, is performed in the same manner as in arithmetic; except that it is usual, in this case, to begin the operation at the left hand, and to proceed towards the right, or contrary to the way of multiplying numbers. The rule is commonly divided into three cases; in each of which, it is necessary to observe, that like signs, in multiplying, produce +, and unlike signs, It is likewise to be remarked, that powers, or roots of C the same quantity, are multiplied together by adding their indices thus, : aXa2, or a1 Xa2=a3; a2 Xa3=a3 ; axa3=a; and am Xanam+n ̧ The multiplication of compound quantities, is also, sometimes, barely denoted by writing them down, with their proper signs, under a vinculum, without performing the whole operation, as 3ab (a-b), or 2a/a+h3. Which method is often preferable to that of executing the entire process, particularly when the product of two or more factors is to be divided by some other quantity, because, in this case, any quantity that is common to both the divisor and dividend, may be more readily suppressed; as will be evident from various instances in the following part of the work. (c) (c) The above rule for the signs may be proved thus: If в, b, be any two quantities, of which в is the greater, and B-b is to be multiplied by a, it is plain that the product, in this case, must be less than aв, because Bb is less than B; and, consequently, when each of the terms of the former are multiplied by a, as above, the result will be For if it were aв + ab, the product would be greater than aв, which is absurd. Also, if в be greater than b, and a greater than a, and it is required to multiply в—b by A — a, the result will be For the product of B-b by a is A (B—¿), or AB—Ab, and that of.B b by a, which is to be taken from the former, is a(B-), as has been already shown; whence Bb being less than B, it is evident that the part which is to be taken away must be less than a ; and consequently since the first part of this product is aB, the second part must be ab; for if it were-ab, a greater part than aв would be to be taken from A(в —b), which is absurd. - CASE I. When the factors are both simple quantities. RULE. Multiply the coefficients of the two terms together, and to the product annex all the letters, or their powers, belonging to each, after the manner of a word; and the result, with the proper sign prefixed, will be the product required (d). (d) When any number of quantities are to be multiplied together, it is the same thing in whatever order they are placed: thus, if ab is to be multiplied by c, the product is either abc, ach, or bca, CASE II. When one of the factors is a compound quantity. RULE. Multiply every term of the compound factor, considered as a multiplicand, separately, by the multiplier, as in the former case; then these products, placed one after another, with their proper signs, will be the whole product required. &c.; though it is usual, in this case, as well as in addition and sub. traction, to put them according to their rank in the alphabet. It may here also be observed, in conformity to the rule given above for the signs, that (+a)x(+b), or (− a)× (—b) = + ab ; and (+a) × (−b), or (−a)×(+b) ab. |