CHAPTER II. THE FOUR FUNDAMENTAL OPERATIONS WITH ADDITION OF ALGEBRAIC NUMBERS. 1. The Addition of two numbers is the process of uniting them into one aggregate. The numbers to be added are called Summands. Addition of Numbers with Like Signs. 2. Ex. 1. Add +3 to +4. The three positive units, +3, when added to the four positive units, +4, give an aggregate of four plus three, or seven, positive units. That is, In like manner, +4 ++3=+(4 + 3) =+7. Ex. 2. -4+3 = ̄(4 + 3) = −7. These examples illustrate the following method of adding two numbers with like signs: Add arithmetically their absolute values, and prefix to the sum their common sign of quality. Addition of Numbers with Unlike Signs. 3. Ex. 1. Add −2 to +5. The two negative units, -2, when added to the five positive units, +5, cancel two of the five positive units. There remain then five minus two, or three, positive units. That is, +5 +−2 =+ (5 − 2) =+3. Ex. 2. Add +2 to −5. The two positive units, +2, when added to the five negative units, -5, cancel two of the five negative units. There remain then five minus two, or three, negative units. That is, −5 ++2 = ̄(5 − 2) = 3. Observe that in both examples the sum is of the same quality as the number which has the greater absolute value. Also, that the absolute value of the sum is obtained by subtracting the less absolute value, 2, from the greater, 5. These examples illustrate the following method of adding two numbers with unlike signs: Subtract arithmetically the less absolute value from the greater. To that remainder prefix the sign of quality of the number which has the greater absolute value. The examples given in Ch. I, Art. 37, are concrete illustrations of the preceding principles. 4. Observe that a positive number increases a number to which it is added, while a negative number decreases it. SUBTRACTION OF ALGEBRAIC NUMBERS. 5. Subtraction is the inverse of addition. In addition two numbers are given, and it is required to find their sum, as In subtraction the sum of two numbers and one of them are given, and it is required to find the other number, as in That is, if from the sum of two numbers either of the numbers be subtracted, the remainder is the other number. 6. Ex. 1. A man's net profits last year were 1200 dollars. This year his income is 150 dollars less, and his expenditures are the same. What are his net profits this year? To take away 150 dollars income is equivalent to adding 150 dollars expenditures. If net profits and income be taken positively, and expenditures negatively, the last statement, expressed algebraically, is +1200 +150 +1200 +-150. = Ex. 2. A man's net profits last year were 1200 dollars. This year his income is the same and his expenditures are 150 dollars less. What are his net profits this year? To take away 150 dollars expenditures is equivalent to adding 150 dollars profits. The algebraic statement of this relation is +1200-150 =+1200 ++150. These examples illustrate the following principle: To subtract one number from another number, reverse the sign of quality of the subtrahend, and add. = E.g., +2+3+2+3, 1. -2 +3 =−2 + ̄3 =−5. -2--3=-2 ++3 = +1. +23+2 ++3, =+5. 7. It is important to notice that the preceding examples do not prove this principle. The following examples illustrate a method of proof which may be used. Ex. 1. Subtract +5 from +7. In +7 +5, the minuend, +7, is to be expressed as the sum of two numbers, one of which is +5. Since 5 +5 = 0, we may write +7=+7 +5 ++5 = (+7 +5)++5. That is, +7 may be regarded as the sum of two numbers, one of which is +7 +-5, and the other is +5. Therefore, by definition of subtraction, +7 − +5 = [(+7 +−5) ++5] −+5 =+7+5=+2, That is, to subtract +5 is equivalent to adding -5. Ex. 2. Subtract -5 from +7. We have +7 −−5 = [(+7 ++5)+ ̄5] −−5 =+7++5=+12, That is, to subtract −5 is equivalent to adding +5. 8. We thus see that every operation of subtraction is equivalent to an operation of addition. On this account it is convenient to speak of a chain of additions and subtractions as an Algebraic Sum. MULTIPLICATION OF ALGEBRAIC NUMBERS. 9. In multiplication, the multiplicand and multiplier are called Factors of the product. 10. In ordinary Arithmetic, multiplication by an integer is defined as an abbreviated addition. Thus, that is, the number 4 is taken three times as a summand. We thus see that the product 4 x 3 is obtained from 4 just as 3 is obtained from the positive unit, 1. We are thus naturally led to the following definition of multiplication: The product is obtained from the multiplicand just as the mul tiplier is obtained from the positive unit. 11. The above definition is an extension of the meaning of arithmetical multiplication when the multiplier is an integer, and gives an intelligible meaning to arithmetical multiplication when the multiplier is a fraction. Thus, is obtained from the unit, 1, by taking one-third of the latter twice as a summand; or In like manner, to multiply 5 by, we take one-third of 5 twice as a summand; or 5 × 3 = 3 + 3 = V. 12. There are two cases to be considered in the multiplication of algebraic numbers. (i.) The Multiplier Positive. Ex. 1. Multiply +4 by +3. By the definition of multiplication, the product, +4 × +3, is obtained from +4 just as +3 is obtained from the positive unit. But +3 =+1 +1 ++1. Consequently the required product is obtained by taking +4 three times as a summand, or +4x+3=+4 ++4 ++4 =+ (4 + 4 + 4) =+ (4 × 3) =+12. |