What is the value of ab, wherein a = m + n and b = p − q, 32. When m= -1, n = +2, p = −3, q = +4 ? 33. When m= +5, n = −6, p = −11, q = -12? 6. The following examples illustrate the meaning of results in the subtraction of algebraic numbers. Ex. 1. Two men, A and B, starting from the same point, P, walk at different rates in the same direction, A 8 miles to the point Q, B 11 miles to the point R. How far is B then from A? As we have seen in Ch. I., distances in one and the same direction may be represented by numbers of the same sign. Let distances toward the right be taken positively, as in Fig. 1, and consequently distances toward the left negatively. The distance of B from A is then represented by QR, and The positive result shows that B is 3 miles to the right of A. In general, however far either may walk, the distance of B from A will always be obtained by subtracting A's distance from the starting point from B's distance from the same point. Ex. 2. If A walks 8 miles to the left and B 11 miles to the left, their distances from P are both negative, as in Fig. 2. The negative result shows that B is 3 miles to the left of A. In a similar way other variations of the problem may be interpreted. EXERCISES IV. 1. A's assets are a dollars and B's are b dollars. What number expresses the excess of A's assets over B's, if assets be taken positively? What number, if assets be taken negatively? What are the meanings of the results of Ex. 1, 2. When a = 3500, b 2750? = 3. When a = 2000, b = 2000 ? 4. When a = 2600, b = 3000? 5. The temperature in Chicago on a certain day was a° and in Philadelphia b. What number expresses the excess of temperature in Chicago. over that in Philadelphia? What is the meaning of the result of Ex. 5, taking temperature above zero positively, The Associative and Commutative Laws for Subtraction. 7. If a number, preceded by the sign+, or the sign, stand first in a chain of additions and subtractions, or first within parentheses, it may be regarded as added to, or subtracted from, 0. Thus, ++8+3=0++8 +3, +3 ++8=0 − +3 ++8. Since every operation of subtraction is equivalent to an operation of addition, it follows that the Associative and Commutative Laws which were proved for addition hold also for subtraction, and for successive additions and subtractions. Ex. = +8+3++8 +3, since +3+3 =+3+8, by Comm. Law Observe that in changing the order of the operations the sign of operation, + or must be transferred with each number. The method of applying the Associative Law depends upon a proper use of parentheses, which will be taken up in the next article. EXERCISES V. Find in three different ways, by applying the Commutative Law, the values of : 1. +83− +4. 4. 2. 17+12 + −5. 3. +28-14 + −2. 31-17 + −36 + +46-11-19 + −49 + +11. 4531 − −15 - +12 + −5 − −9+ −8 + +4. Find, in the most convenient way, the values of : 6. 103-12 − +3. 7. -799-11 + −1. Removal of Parentheses. 8. We have +9 + (+5++6) =+9++5++6, since to add the sum +5 ++6 is equivalent to adding successively the single numbers of that sum. Again, +9 +(+5 − +6) = +9 + (+5 +−6), since — +6 = −6, +9++5+6, removing parentheses, +9++56, since +6=—+6. = This example illustrates the following principle: (i.) When the sign of addition, +, precedes parentheses, they may be removed, and the signs of operation, + and —, within them be left unchanged; that is, We have N+ (+ a + b) = N + a + b, N+(+a-5)=N+ab, etc. +9 (56) =+9 +5 +6, since to subtract the sum 5++6 is equivalent to subtracting successively the single numbers of that sum. Again, +9 (+5 − +6) = +9 − (+5 +6), since +6 = + ̄6, - This example illustrates the following principle: (ii.) When the sign of subtraction, -, precedes parentheses, they may be removed, if the signs of operation within them be reversed from + to, and from to; that is, N+(+a++b) = N + +a + +b, since to add the sum +a++b is equivalent to adding successively the single numbers of that sum. N+ (+a − +b) = N + (+a + −b), since − +b =+ −b, = N++a + −b, removing parentheses, = N++a − +b, since + b = − +b. Evidently the preceding proof does not depend upon the signs of quality of the numbers within the parentheses, nor upon how many numbers are inclosed. In a similar manner (ii.) is proved. Insertion of Parentheses. 9. The insertion of parentheses is the converse of the process of removing them. (i.) An expression may be inclosed within parentheses preceded by the sign of addition, if the signs of operation, + and -, preceding the numbers inclosed within the parentheses remain unchanged. E.g., ++7 +5 +−3 − ̄4 = ++7 + ( − +5 +−3 −−4) = ++7 −+5 + (+ ̄3 − ̄4) (ii.) An expression may be inclosed within parentheses preceded by the sign of subtraction, if the signs of operation preceding the numbers inclosed within the parentheses be reversed, from + to and from E.g., to +. = ++7 +5 +34 ++7 − ( ++5 −3+4) - The insertion of parentheses is a direct application of the Associative Law. EXERCISES VI. Find the values of the following expressions, first removing parentheses: Insert parentheses in the expression +8 -5 +−3 − +7, 11. To inclose the last three numbers preceded by the sign +; preceded by the sign 12. To inclose the last two numbers preceded by the sign+; preceded by the sign 13. To inclose the first and third numbers preceded by the sign + ; preceded by the sign -. 10. In ordinary Arithmetic, to subtract a number from any number decreases the latter. But such is not always the case in subtracting one algebraic number from another. Thus, +7+4 = +3, and +3 <+7; but +7411, and +11 > +7. Property of Zero in Subtraction. 11. From § 1, Art. 13, we have N+0= N. If, therefore, from N, which is the sum of N and 0, be subtracted either N or 0, the remainder is 0 or N, respectively, by the definition of subtraction. That is, N-NO, and N-0=N. § 3. MULTIPLICATION OF ALGEBRAIC NUMBERS. (i.) 1. As in Arithmetic, the number multiplied is called the Multiplicand, the number that multiplies the Multiplier, and the result the Product. In ordinary Arithmetic, multiplication by an integer is defined as an abbreviated addition. Thus, to |