ence to sign, it is equivalent to regarding them of the same sign. Algebraic Subtraction shows the method of finding the difference between two quantities which have either the same or unlike signs; and it frequently happens, that this difference is greater than either of the quantities. To understand this properly, requires a knowledge of the nature of positive and negative quantities. All quantities are to be regarded as positive, unless, for some special reason, they are otherwise designated. Negative quantities embrace those that are, in their nature, the opposite of positive quantities. Thus, if a merchant's gains are positive, his losses are negative; if latitude north of the equator is reckoned +, that south, would be; if distance to the right of a certain line is reckoned +, then distance to the left would be -; if elevation above a certain point, or plane, is regarded as +, then distance below would be -; if time after a certain hour is +, then time before that hour is —; if motion in one direction is, then motion in an opposite direction would be - ; and so on. With this knowledge of the meaning of the sign minus, it is easy to see how the difference of two quantities having the same sign, is equal to their difference; and also, how the difference of two quantities having different signs, is equal to their sum. 1. One place is situated 10, and another 6 degrees north of the equator, what is their difference of latitude? Here we are required to find the difference between +10 and +6, which is evidently +4; by which we are to understand that the first place is 4 degrees farther north than the second. 2. Two places are situated, one in 10, and the other in 6 degrees south latitude; what is the difference of latitude? Here we are required to find the difference between -6 and -10, which is evidently -4, by which we learn, that the first place is 4 degrees farther south than the second. 3. One place is situated in 10 degrees north, and another in 6 degrees south latitude; what is their difference of latitude? Here we are required to find the difference between +10 and -6, or to take -—6 from +10, which, by the rule for subtraction, leaves +16; which is evidently the difference of their latitudes, and from which we learn, that the first place is 16 degrees farther north than the other. It is thus, when properly understood, the results are always capable of a satisfactory explanation. REVIEW.-64. In what respects does algebraic differ from arithmetical Subtraction? In what respect do negative quantities differ from positive? Illustrate the difference by examples. MULTIPLICATION. ART. 65. MULTIPLICATION, in Algebra, is the process of taking one algebraic expression, as often as there are units in another. In Algebra, as in Arithmetic, the quantity to be multiplied is called the multiplicand; the quantity by which we multiply, the multiplier, and the result of the operation, the product. The multiplicand and multiplier are generally called factors. ART. 66. Since the quantity a, taken once, is represented by a, when taken twice, by a+a, or 2a, when taken three times, by a+a+a, or 3a, it is evident, that to multiply a literal quantity by a number, it is only necessary to write the multiplier as the coefficient of the literal quantity. 1. If 1 lemon costs a cents, how many cents will 5 lemons cost? If one lemon costs a cents, five lemons will cost five times as much, that is 5a cents. 2. If 1 orange costs c cents, how many cents will 6 oranges cost? 3. A merchant bought a pieces of cloth, each containing b yards, at c dollars per yard; how many dollars did the whole cost? In a pieces, the number of yards would be represented by ab, or ba, and the cost of ab yards at c dollars per yard, would be represented by c taken ab times, that is, by abc, which is represented by abc. ART. 67. It is shown in "Ray's Arithmetic," Part III, Art. 44, that the product of two factors is the same, whichever be made the multiplier; we will, however, demonstrate the principle here. Suppose we have a sash containing a vertical, and b horizontal rows; there will be a panes in each horizontal row, and b panes in each vertical row; it is required to find the number of panes in the window. It is evident, that the whole number of panes in the window will be equal to the number in one row, taken as many times as there are rows. Then, since there are a vertical rows, and b panes in each row, the whole number of panes will be represented by b taken a times, that is, by ab. Again, since there are b horizontal rows, and a panes in each row, the whole number of panes will be represented by a taken b times, that is, by ba. But, since either of the expressions, ba or REVIEW.-65. What is Multiplication in Algebra? What is the multiplicand? The multiplier? The product? What are the multiplicand and multiplier generally called? 66. How do you multiply a literal quantity by a number? ab, represents the whole number of panes in the window, they are equal to each other, that is, ab is equal to ba. Hence, it follows, that the product of two factors is the same, whichever be made the multiplier. By taking a 3 and b-4, the figure in the margin may be used to illustrate the principle in a particular case. In the same manner, the product of three or more quantities is the same, in whatever order they are taken. Thus, 2×3×4=3×2×4=4×2×3, since the product in each case is 24. 1. What will 2 boxes, each containing a lemons, cost at b cents per lemon? One box will cost ab cents, and 2 boxes will cost twice as much as 1 box, that is, 2ab cents. 2. What is the product of 2b, multiplied by 3a? The product will be represented by 26×3a, or by 3aX2b, or by 2×3×ab, since the product is the same, in whatever order the factors are placed. But 2×3 is equal to 6, hence the product 2b3a is equal to Gab. Hence, we see, that in multiplying one monomial by another, the coefficient of the product is obtained by multiplying together the coefficients of the multiplicand and multiplier. This is termed, the rule of the coefficients. ART. 68. Since the product of two or more factors is the same, in whatever order they are written, if we take the product of any two factors, as 2×3, and multiply it by any number, as 5, the product may be written 5×2×3, or 5×3×2, that is, 10X3, or 15×2, either of which is equal to 30. From which we see, that. when either of the factors of a product is multiplied, the product itself is multiplied. ART. 69.-1. What is the product of a by a? The product of b by a is written ab, hence, the product of a by a would be written aa; but this, (Art. 33,) for the sake of brevity, is written a2. 2. What is the product of a2 by a? Since a2 may be written thus, aa, the product of a2 by a may be REVIEW.-67. Prove that 3 times 4 is the same as 4 times 3. Prove that a times b is the same as b times a. Is the product of any number of factors changed by altering their arrangement? In multiplying one monomial by another, how is the coefficient of the product obtained? 68. If you multiply one of the factors of a product, how does it affect the product? 69. How may the product of a by a be written? How may the product of as by a be written? expressed thus, aa×a, or aaa, which, for the sake of brevity, is written a3. Hence, the exponent of a letter in the product, is equal to the sum of its exponents in the two factors. rule of the exponents. 3. What is the product of a2 by a2? This is termed, the Ans. aaaa, or a*. 4. What is the product of a2b by ab? Ans. aaabb, or a3b2. • 5. What is the product of 2ab2 by 3ab? Ans. 6aabbb, or 6a2b3 Hence, the RULE, FOR MULTIPLYING ONE POSITIVE MONOMIAL BY ANOTHER. Multiply the coefficients of the two terms together, and to their product annex all the letters in both quantities, giving to each letter an exponent equal to the sum of its exponents in the two factors. NOTE. It is customary to write the letters in the order of the alphabet. Thus, abXc is generally written abc. 14. What is the product of 3a3b2c by 5ab2c3 ? 15. What is the product of 7xyz by 8x3yz? NOTE. The learner must be careful to distinguish between the coefficient and the exponent. Thus, 2a is different from a2. To fix this in his mind, let him answer such questions as the following: What is 2a-a2 equal to, when a is 1? Ans. 1. ART. 70.-1. Suppose you purchase 5 oranges at 4 cents a piece, and pay for them, and then purchase 2 lemons at the same price; what will be the cost of the whole? 5 oranges, at 4 cents each, will cost 20 cents; 2 lemons, at 4 cents each, will cost 8 cents, and the cost of the whole will be 20+8=28 cents. The work may be written thus: 5+2 4 20+8=28 cents. If you purchase a oranges at c cents a piece, and b lemons at c cents a piece, what will be the cost of the whole? The cost of a oranges, at c cents each, will be ac cents; the cost of b lemons, at c cents each, will be bc cents, and the whole cost will be ac+bc cents. The work may be written thus: a+b с ac+bc Hence, when the sign of each term is positive, we have the following RULE, FOR MULTIPLYING A POLYNOMIAL BY A MONOMIAL. Multiply each term of the multiplicand by the multiplier. EXAMPLES. 2. Multiply a+d by b. 3. Multiply ac+be by d. 4. Multiply 4x+5y by 3a. 5. Multiply 2x+3z by 2b. 6. Multiply m+2n by 3n. 7. Multiply x+y by ax. 8. Multiply +y2 by xy.. 9. Multiply 2x+5y by abx. 10. Multiply 3x2+2xz by 2xz. 11. Multiply 3a+2b+5c by 4d.. 12. Multiply be+af+mx by 3ax. 13. Multiply ab+ax+xy by abxy.. Ans. 6x+4x2z2. Ans. 12ad+8bd+20cd. Ans. 3abcx+3a2ƒx+3amx2. Ans. a2b2xy+a2bx2y+abx2y2. find the product of r+y by ART. 71.—1. Let it be required to a+b. Here the multiplicand is to be taken as many times as there are units in a+b, and the whole product will evidently be equal to the sum of the two partial products. Thus, x+y ax+ay=the multiplicand taken a times. bx+by=the multiplicand taken b times. ax+ay+bx+by the multiplicand taken (a+b) times. If x=5, y=6, a=2, and b=3, the multiplication may be arranged thus: 5+6 2+3 10+12=the multiplicand taken 2 times. 15+18 the multiplicand taken 3 times. |