By a simple process the values of x and y are readily determined. From this parallel between arithmetical and algebraic forms of expression the brevity and the advantage of the literal or general symbo for number is clearly manifest. The later processes of algebra will constantly furnish the means wherewith we may broaden our power of expression, and the meaning of algebra will be interpreted as merely an extension of our processes with number. THE SYMBOLS OF OPERATION 5. The principal signs for operations in algebra are identical with those of the corresponding operations in arithmetic. 6. Addition is indicated by the "plus" sign, +. Thus, a + b is the indicated sum of the quantity a and the quantity b. The expression is read "a plus b." 7. Subtraction is indicated by the "minus" sign, Thus, a − b is the indicated difference between the quantity b and the quantity a. 'The expression is read "a minus b." 8. Multiplication is usually indicated by an absence of sign between the quantities to be multiplied. Thus, ab is the indicated product of the quantities a and b. abx is the indicated product of the quantities a, b, and î Sometimes a dot is used to indicate a multiplication. Thus, a. b is the product of a and b. The ordinary symbol, "x," is occasionally used in algebraic expression. An indicated product may be read by the use of the word "times" or by reading the literal symbols only. Thus, ab may be read "a times b," or simply "ab." 9. Division is indicated by the sign "+," or by writing in the fractional form. Thus, ab is the indicated quotient of the quantity a divided by the quantity b. a b is the fractional form for the same indicated quotient. Both forms are read "a divided by b." 10. Indicated operations are of constant occurrence in algebraic processes, for the literal symbols do not permit the combining of two or more into a single symbol as in the case of numerals. Thus: Arithmetically, 5+3+ 7 may be written "15," for the symbol 15 is the symbol for the group made up of the three groups, 5, 3, and 7. Algebraically, a + b + c cannot be rewritten unless particular values are assigned to the symbols a, b, and c. The sum is an indicated result. Algebraic expression, therefore, confines us to a constant use of indicated operations, and we must clearly understand the meaning of: 11. Equality of quantities or expressions in algebra is indicated by the sign of equality, =, read "equals," or "is equal to." is an indicated equality between two quantities, a + b and c + d. 12. If two or more numbers are multiplied together, each of them, or the product of two or more of them, is a factor of the product; and any factor of a product may be considered the coefficient of the product of the other factors. Thus: In 5 a, 5 is the coefficient of a. or x is the coefficient of a. In acmx, a is the coefficient of cmx, or ac may be the coefficient of mx, etc. Coefficients are the direct results of additions, for 5 a is merely an abbreviation of a + a + a + a + a. If the coefficient of a quantity is "unity" or "1," it is not usually written or read. Thus, a is the same as 1 a. xy is the same as 1 ry. 13. The parenthesis is used to indicate that two or more quantities are to be treated as a single quantity. The ordinary form, (), is most common. For clearness in the discussion of elementary principles, the parenthesis will frequently be made use of to inclose single quantities. 14. An axiom is a statement of a truth so simple as to be accepted without proof. Two of the axioms necessary in early discussions are: AXIOM 1. If equals are added to equals, the sums are equal. AXIOM 2. If equals are subtracted from equals, the remainders are equal. THE SYMBOLS OF QUALITY 15. In scientific and in many everyday discussions greater clearness and convenience have resulted from a definite method of indicating opposition of quality. For example: Temperature above and below the zero point, Assets, or possessions, and liabilities, or debts, in business, etc., represent cases in which direct opposites of quality or kind are under discussion; hence, a need exists for a form of expression that shall indicate kind as well as amount, quality as well as quantity. To supply this need the plus and minus signs are in general use, their direct opposition making them useful as signs of quality as well as of operation; and we will now consider their use as 16. A few selected examples of common occurrence clearly illustrate the application of quality signs to opposites in kind In the laboratory: Temperature above 0° is considered as +. In navigation: Latitude north of the equator is considered as +. In business administration: Assets, or possessions, are considered as +. signs The following illustration emphasizes the advantage of indicating opposites in kind by the use of the + and of quality. A thermometer registers 10° above 0 at 8 A.M., 15° above 0 at 11 A.M., 5° below 0 at 4 P.M., 10° below at 10 P.M. At the right we have tabulated the conditions in a concise form made possible only through the use of quality signs. and By applying this idea of opposition to arithmetical numbers we may establish NEGATIVE NUMBERS 17. It is first necessary to show that there exists a need for a definite method of indicating opposition of quality in number. Consider the subtractions: (1) 5-4= (2) 5 −5 = There are three definite cases included. (3) 5-6= In (1) a subtrahend less than the minuend. The first two cases are familiar in arithmetical processes, but the third raises a new question, and we ask, 18. On any convenient straight line denote the middle point by "0,” and mark off equal points of division both to ⚫ the right and to the left of 0. The two directions from 0 are clearly defined cases of opposition, and this opposition may be indicated by marking the suc cessive division points at the right of 0 with numerals having plus signs, -5-4-3-2 -1 +1 +2 +3 +4 +5 and, similarly, the division points left of 0 with numerals having minus signs. The result is a series of positive and negative numbers established from a given point which we may call "zero." With this extended number system we may at once obtain a clear and logical answer for the question raised above. The minuend remaining the same in each case, the result for each subtraction is established by merely counting off the subtrahend from the minuend, the direction of counting being toward zero. Therefore, For (1) The subtraction of a positive number from a greater positive number gives a positive result. For (2) The subtraction of a positive number from an equal positive number gives a zero result. For (3) The subtraction of a positive number from a less positive number gives a negative result. |