SEOTION 111. ALGEBRA, CHAPTER I. ELEMENTARY PRINCIPLES. 1. Algebra treats of numbers, the numbers being represented by letters (symbols of quantity), affected with certain symbols of quality, and connected by symbols of operation. It is easy to see that these symbols of quantity may be dealt with very much as we deal with concrete quantities in arithmetic. Thus, allowing the letter a to stand for the number of units in any quantity, and allowing also 2 a, 3 a, 4 a, &c., to stand respectively for twice, thrice, four times, &c., as large a quantity as the letter a, it at once follows that we may perform the operations of addition, subtraction, multiplication, and division upon these symbols exactly as we do in ordinary arithmetic upon concrete quantities. For instance, 4 a and 6 a make 10 a, 9 a exceeds 5 a by 4 a, 15 a is 5 times 3 a, and 7 a is contained 8 times in 56 a. Neither is it necessary in these operations to state, or even to know the exact number of units for which any symbol of quantity stands, nor indeed the nature of these units ; it is simply sufficient that it is a symbol of quantity. Thus, in the science of chemistry, we use a weight called a crith ; and a person unacquainted with chemistry might not know wbether a crith were a measure of lengtlı, weight, or capacity, or indeed whether it were a measure at all, yet he would at once allow that 6 criths and 5 criths are 11 criths, that twice 4 criths are 8 criths, &c: The Signs + and as Symbols of Operation. 2. In purely arithmetical operations, the signs + and are respectively the signs of addition and subtraction. In this sense, too, they are used in algebra. Thus, a + b means that 6 is to be added to a, and a – 6 means that b is to be subtracted from a. Hence, as long as a and b represent ordinary arithmetical numbers, a + b admits of easy interpretation, as also does a – b, when b is not greater than a. But when is greater than a, the expression a - 6 has no arithmetical meaning, By an extension, however, of the use of the signs + and -, we are able to give such expressions an intelligible signification, whatever may be the quantities represented by a and b. with a Positive and Negative Quantities.-The Signs + and as Symbols of Affection or Quality. 3. DEF.—A positive quantity is one which is affected + sign, and a negative quantity is one which is affected with a · sign. Let BA be a straight line, and 0 a point in the line; and suppose a person, starting from 0, to walk a miles in the direction OA. Suppose also another person, starting from the same or any other point in BA to walk a miles in the direction OB. These persons will thus walk a miles each in exactly opposite directions. Now, we call one of these directions positive (it matters not which) and the other negative. Let us take the direction OA as positive. We then have the first person walking a miles in a positive direction, and the second walking a miles in a negative direction. We represent these distances algebraically by + a and respectively. It will therefore be seen that the signs + and effect upon the magnitudes of quantities, but that they express the quality or affection of the quantities before which they stand. a have no Again, suppose a person in business to get a profit of £6, while another suffers a loss of £6. We may express these facts algebraically in two ways. We may consider gain as positive, and loss as negative gain, and say that the former has gained + 6 pounds, while the latter has gained - 6 pounds. Or we may consider loss as positive, and gain as negative loss, and say that the former has lost – 6 pounds, while the latter has lost + 6 pounds. We hence see that the gain of + 6 is equal to a loss of 6, and that a gain of 6 is equal to a loss of + 6. B X The Sum of Algebraical Quantities. 4. Let a distance AB be measured to the right along the line AX. And let a further distance BC be measured from B in the same direction. By the sum of these lines we mean the resulting distance of the point C from the original point A, that is to say, the distance AC. (It may be remarked that we add the line BC to the line AB by measuring BC in its own proper direction from the extremity B of AB. It is hardly necessary to remind the student that both lines are in the same straight line AX.) Let us represent the distances AB and BC by + a and + b respectively; then the algebraical sum of the lines will be represented writing these quantities side by side, each with its own proper sign of affection. Thus the sum of the distances AB and BC is expressed by + a + b, or, as it is usual to omit the + sign of a positive quantity when the quantity stands alone or at the head of an algebraical expression, the sum of AB and BC is expressed by a + b. Hence, the interpretation of a + b is that it represents the distance AC. Again, taking as above + a to represent the distance AB along the straight line AX, and & measured to the right, let a distance BC be measured from B in the same straight line AX, but this time to the left. A B X AC B B X Let the latter distance be represented by - b. Then, on the principle above, AC is the sum of these distances, and this sum is represented algebraically by + Q - b or a - b. (It will be seen that the distance BC is again measured from B in its own proper direction, and that the resultant distance AC of the point C is again the sum of the line AB and BC.) There will evidently be three cases, viz. : 1. When the distance BC is less in magnitude than AB, in which case the point C is on the right of A, and the distance AC is positive. 2. When the distance BC is equal in magnitude to AB, in which case the point C coincides with A, and the distance AC is zero. 3. When the distance BC is greater in magnitude than AB, the point C being then on the left of A, and the distance is negative. Now, a 6 in all these cases represents the distance AC. It therefore admits of intelligible interpretation whether b be less than, equal to, or greater than a. And, since the distance AC is obtained in the first two cases by subtracting the distance BC from that of AB, and in the second case by subtracting as far as AB will allow of subtraction, and measuring the remainder to be subtracted in an opposite direction, it follows that The sign -, which, standing before a letter, is a symbol of quality, becomes at once a symbol of subtraction in all cases when the quantity in question is placed immediately after any other given quantity with its proper sign of affection. Hence also we may conclude that the addition of a negative quantity is equivalent to the subtraction of the corresponding positive quantity. 5. We may prove in a similar way thatThe subtraction of a negative quantity is equivalent to the addition of the corresponding positive quantity. Let, as before, + a represent the distance AB, measured from the point A to the right, and let it be required to subtract from + a the distance represented by b. A B Now, in the last article, we added a distance to a given distance AB, by measuring the second distance in its own direction from the extremity B. We shall therefore be consistent if we subtract a given distance – 6 by measuring this distance in a direction exactly opposite to its own direction, from the same extremity B. Now, the direction of -b is to the left. If, therefore, we measure a distance BC to the right, equal in magnitude to the distance to be subtracted, we obtain a distance AC which is correctly represented by a -(-). But AC is also correctly represented by a + b, and hence it follows that Q - (- 6) = a + b. We may apply the above principle to all magnitudes which admit of continuous and indefinite extension; as, for instance, to forces which pull and push, attract and repel; to time past and time to come, to temperatures above zero and below zero, to money due and money owed, to distance up and distance down, &c., in all which cases, having represented one by a quantity affected with a + sign, we may represent the other by a quantity affected with a - sign. 6. In expressing the sum of a number of quantities, the order of the terms is immaterial. We will take, as our illustration, a body subject to various alterations of temperature, and we will suppose the temperature of the body, before the changes in question, to be zero or 0°. Let the temperature now undergo the following changesviz., a rise of a', a fall of bo, a fall of c', and, lastly, a rise of do. Let us consider a rise as positive, and therefore a fall as negative. We may then represent these changes respectively by + a, - b, -C, + d. And it is further evident that the resulting temperature will be represented by the sum of these quantities, which, as previously written, will be a 6 But again, it is plain that the resulting temperature of the body will not be affected if these changes of temperature take place in the reverse order, or in any other order. Thus, suppose the temperature first falls c', then rises ao, then rises alo, and, lastly, falls b°, it is evident that the final temperature will be the same as before. And the sum of the quantities C, + a, + d, – b, represents this final temperature. Ct d. |