ELEMENTARY ALGEBRA. 1. I. ADDITION AND SUBTRACTION. ALGEBRA is the science which teaches the use of sYMBOLS to denote numbers and the operations to which numbers may be subjected. 2. The symbols employed in Algebra to denote numbers are, in addition to those of Arithmetic, the letters of some alphabet. Thus a, b, c ...... x, y, z: a, ß, y ......: d', b', c' ...... read ...... 3. The number one, or unity, is taken as the foundation of all numbers, and all other numbers are derived from it by the process of addition. Thus two is defined to be the number that results from adding one to one; three is defined to be the number that results from adding one to two; four is defined to be the number that results from adding one to three; and so on. 4. The symbol +, read plus, is used to denote the opera tion of ADDITION. and Thus 1+1 symbolizes that which is denoted by 2, 2+1 3, a+b stands for the result obtained by adding b to a. 5. The symbol stands for the words "is equal to," or "the result is." 8. A. 1 Thus the definitions given in Art. 3 may be presented in an algebraical form thus: 2=1+1, where unity is written twice, 3=2+1=1+1+1, where unity is written three times, 4=3+1=1+1+1+1 it follows that a=1+1+1 four times, +1+1 with unity written a times, b=1+1+1 +1+1 with unity written b times. ...... 7. The process of addition in Arithmetic can be presented in a shorter form by the use of the sign +. Thus if we have to add 14, 17, and 23 together we can represent the process thus: 14+17+23 = 54. 8. When several numbers are added together it is indifferent in what order the numbers are taken. Thus if 14, 17, and 23 be added together their sum will be the same in whatever order they be set down in the common Arithmetical process: So also in Algebra when any number of symbols are added together, the result will be the same in whatever order the symbols succeed each other. Thus if we have to add together the numbers symbolized by a and b, the result is represented by a+b, and this result is the same number as that which is represented by b+a. Similarly the result obtained by adding together a, b, c might be expressed algebraically by a+b+c, or a+c+b, or b+a+c, or b+c+a, or c+a+b, or c+b+a. 9. When a number denoted by a is added to itself, the result is represented algebraically by a+a. This result is for the sake of brevity represented by 2a, the figure prefixed to the symbol expressing the number of times the number denoted by a is repeated. Similarly a+a+a is represented by 3a. Hence it follows that 10. The symbol —, read minus, is used to denote the operation of SUBTRACTION. Thus the operation of subtracting 15 from 26 and its conection with the result may be briefly expressed thus; 26-15=11. 11. The result of subtracting the number b from the number a is represented by a-b. Again a-b-c stands for the number obtained by taking c from a-b. Also a-b-c-d stands for the number obtained by taking d from a-b-c. Since we cannot take away a greater number from a smaller, the expression a-b, where a and b represent numbers, can denote a possible result only when a is not less than b. So also the expression a-b-c can denote a possible result only when the number obtained by taking b from a is not less than c. 12. A combination of symbols is termed an algebraical expression. The parts of an expression which are connected by the symbols of operation + and are called TERMS. Compound expressions are those which have more than one term. Thus a−b+c-d is a compound expression made up of four terms. When a compound expression contains two terms it is called a Binomial, Terms which are preceded by the symbol + are called positive terms. Terms which are preceded by the symbol are called negative terms. When no symbol precedes a term the symbol is understood. Thus in the expression a-b+c−d+e-f a, c, e are called positive terms, The symbols of operation + and - are usually called positive and negative SIGNS. 13. If the number 6 be added to the number 13, and if 6 be taken from the result, the final result will plainly be 13. So also if a number b be added to a number a, and if b be taken from the result, the final result will be a: that is, a+b-b=a. Since the operations of addition and subtraction when performed by the same number neutralize each other, we conclude that we may obliterate the same symbol when it presents itself as a positive term and also as a negative term in the same expression. 14. If we have to add the numbers 54, 17, and 23, we may first add 17 and 23, and add their sum 40 to the number 54, thus obtaining the final result 94. This process may be represented Algebraically by enclosing 17 and 23 in a BRACKET ( ), thus: 54+ (17+23)=54+40=94. 15. If we have to subtract from 54 the sum of 17 and 23, the process may be represented Algebraically thus: 54-(17+23)=54-40=14. 16. If we have to add to 54 the difference between 23 and 17, the process may be represented Algebraically thus: 54+(23-17)=54+6=60. 17. If we have to subtract from 54 the difference between 23 and 17, the process may be represented Algebraically thus: 54-(23-17)=54-6=48. 18. The use of brackets is so frequent in Algebra, that the rules for their removal and introduction must be carefully considered. We shall first treat of the removal of brackets in cases where symbols supply the places of numbers corresponding to the Arithmetical examples considered in Arts. 14, 15, 16, 17. CASE I. To add to a the sum of b and c. First add b to a, the result will be a+b. This result is too small, for we have to add to a a number greater than b, and greater by c. Hence our final result will be obtained by adding c to a+b, and it will be a+b+c. CASE II. To take from a the sum of b and c. This is expressed thus: a- (b+c). First take b from a, the result will be a-b. This result is too large, for we have to take from a a number greater than b, and greater by c. Hence our final result will be obtained by taking c from a-b, and it will be a-b-c. CASE III. To add to a the difference between b and c. This is expressed thus: a+ (b−c). First add b to a, the result will be a+b. This result is too large, for we have to add to a a number less than b, and less by c. Hence our final result will be obtained by taking c from a+b, and it will be a+b-c. CASE IV. To take from a the difference between b and c. This is expressed thus: a- - (b −c). First take b from a, the result will be a-b. This result is too small, for we have to také from a a number less than b, and less by c. Hence our final result will be obtained by adding c to a-b, and it will be a-b+c. |
