Art. 23. ALGEBRA may be defined, a general method of investigating the relations of quantities, by letters, and other symbols. This, it must be acknowledged, is an imperfect account of the subject; as every account must necessarily be, which is comprised in the compass of a definition. Its real nature is to be learned, rather by an attentive examination of its parts, than from any summary description.
The solutions in Algebra, are of a more general nature than those in common Arithmetic. Tne latter relate to particular numbers; the former to whole classes of quantities. On this account, Algebra has been termed a kmd of universal Arithmetic. The generality of its solutions is principally owing to the use of letters, instead of numeral figures, to express the several quantities which are subjected to calculation. In Arithmetic, when a problem is solved, the answer is limited to the particular numbers which are specified, in the statement of the question. But an Algebraic solution may be equally applicable to all other quantities which have the same relations. This important advantage is owing to the difference between the customary use of figures, and the manner in which letters are employed in One of the nine digits, invariably expresses the same number: but a letter may be put for any number whatever. The figure 8 always signifies eight; the figure 5, five, &c. And, though one of the digits, in connection with others, may have a local value, different from its simple value when afone; yet the same combination always expresses the same number. Thus 263 has one uniform signification. And this is the case with every other combination of figures. But in Algebra, a letter may stand for any quantity which we wish it to represent. Thus b may be put for 2, or 10, or 50, or 1000. It must not be understood from this, however, that the letter has no determinate value. Its value is fixed for the occasion. For the present purpose, it remains unaltered. But on a different occasion, the some letter may be put for any other number.
A calculation may be greatly abridged by the use of letters; especially when very large numbers are concerned. And when several such numbers are to be combined, as ip multiplication, the process becomes extremely tedious. But a single letter may be put for a large number, as well as for a small one. The numbers 26347297, 68347823, and 27462498, for instance, may be expressed by the letters, b, c, and d. The multiplying them together, as will be seen hereafter, will be nothing more than writing them, one after another, in the form of a word, and the product will be simply bcd. Thus in Algebra, much of the labor of calculation may be saved, by the rapidity of the operations. Solutions are sometimes effected, in the compass of a few lines, which, in common Aritlnnetic, must be extended through many pages.
24. Another advantage obtained from the notation by letters instead of figures, is, that the several quantities which are brought into calculation, may be preserved distinct from each other; though carried through a number of complicated processes; whereas, in arithmetic they are so blended together, that no trace is left of what they were, before the operation began.
25. Algebra differs farther from arithmetic, in making use of unknown quantities, in carrying on its operations. In arithmetic, all the quantities which enter into a calculation must be known. For they are expressed in numbers. And every number must necessarily be a determinate quantity. But in Algebra, a letter may be put for a quantity, before its value has been ascertained. And yet it may have such relations to other quantities, with which it is connected, as to answer an important purpose in the calculation.
26. To facilitate the investigations in algebra, the several steps of the reasoning, instead of being expressed in words, are translated into the language of signs and symbols, which may be considered as a species of short-hand. This serves to place the quantities and their relations distinctly before the eye, and to bring them all into view at once. They are thus more readily compared and understood, than when removed at a distance from each other, as in the common mode of writing. But before any one can avail himself of this advantage, he must become perfectly familiar with the new language.
27. The quantities in algebra, as has been already observed, are generally expressed by letters. The first letters of the Alphabet are used to represent known quantities; and the last letters, those which are unknown. Sometimes the quantities, instead of being expressed by letters, are set down in figures, as in common arithmetic.
28. Besides the letters and figures, there are certain characters used, to indicate the relations of the quantities, or the operations which are performed with them. Among these are the signs + and -, which are read plus and minus, or more and less. The fonner is prefixed to quantities which are to be added; the latter, to those which are to be subtracted. Thus a + b signifies that b is to be added to a. It is read a plus b, or a added to b, or a and b. If the expression be a - b, i. e. a minus b; it indicates that b is to be subtracted from a.
29. The sign + is prefixed to quantities which are considered as affirmative or positive; and the sign -, to those which are supposed to be negative.
All the quantities which enter into an algebraic process, are considered, for the purposes of calculation, as either positive or negative. Before the first one, unless it be negative, the sign is generally omitted. But it is always to be understood. Thus a + b, is the same as +a + b.
30. Sometimes both + and - are prefixed to the same letter. The sign is then said to be ambiguous. Thus a ± b signifies that in certain cases, compiehended in a general solution, b is to be added to a, and in other cases subtracted from it.
31. The equality between two quantities or sets of quantities is expressed by parallel lines =. Thus a + b = d signifies that a and b together are equal to d. And a + d = c = b + g = h signifies that a and d equal c, which is equal to b and g, which are equal to h. So 8 + 4 = 16 - 4 = 10 + 2 = 7 + 2 + 3 = 12.
32. When the first of the two quantities compared, is greater than the other, the character > is placed between them. Thus a > b signifies that a is greater than b.
If the first is less than the other, the character < is used; as a < b; i.e. a is less than b. In both cases, the quantity towards which the character opens, is greater than the other.
33. A numeral figure is often prefixed to a letter. This is called a coefficient. It shows how often the quantity expressed by the letter is to be taken. Thus 2b signifies twice b; and 9b, 9 times 6, or 9 multiplied into b.
The coefficient may be either a whole number or a fraction. Thus (2/3)b is two-thirds of b. When the coefficient is not expressed, 1 is always to be understood. Thus a is the same as 1a; i. e. once a.
34. The coefficient may be a letter, as well as a figure. In the quantity mb, m may be considered the coefficient of b; because b is to be taken as many times as there are units in m. If m stands for 6, then mb is 6 times b. In 3abc, 3 may be considered as the coefficient of abc; 3a the coefficient of bc; or 3ab, the coefficient of c.
35. A simple quantity is either a single letter or number, or several letters connected together without the signs + and -. Thus a, ab, abd and 8b are each of them simple quantities. A compound quantity consists of a number of simple quantities connected by the sign + or -. Thus a + b, d - y, b - d + 3h, are each compound quantities. The members of which it is composed are called terms.
36. If there are two terms in a compound quantity, it is called a binomial. Thus a + b and a - b are binomials. The latter is also called a residual quantity, because it expresses the difference of two quantities, or the remainder, after one is taken from the other. A compound quantity consisting of three terms, is sometimes called a trinomial; one of four terms, a quadrinomial, &c.
37. When the several members of a compound quantity are to be subjected to the same operation, they are frequently connected by a line called a vinculum. Thus a - (b + c) shows that the sum of b and c is to be subtracted from a. But a - b + c signifies that b only is to be subtracted from a while c is to be added. The sum of c and d, subtracted from the sum of a and b, is (a + b) - (c + d). The equality of two sets of quantities is expressed, without using a vinculum. Thus a + b = c + d signifies, not that b is equal to c; but that the sum of a and b is equal to the sum of c and d.
38. A single letter, or a number of letters, representing any quantities with their relations, is called an algebraic expression; and sometimes a formula. Thus a + b + 3d is an algebraic expression.
39. The character . denotes multiplication. Thus a.b is a multiplied into b: and 6.3 is 6 times 3, or 6 into 3. Sometimes a sign x is used to indicate multiplication. Thus a x b is the same as a.b. But the point is more commonly omitted, between simple quantities; and the letters are connected together, in the form of a word or syllable. Thus ab is the same as a.b or a x b. And bcde is the same as b.c.d.e. When a compound quantity is to be multiplied, paretheses are used, as in the case of subtraction. Thus the sum of a and b multiplied into the sum of c and d, is (a + b).(c + d). And (6 + 2).5 is 8.5 or 40. But 6 + 2.5 is 6 + 10 or 16. When the parentheses are used, the sign of multiplication is frequently omitted. Thus (x + y)(x - y) is (x + y).(x - y).
40. When two or more quantities are multiplied together, each of them is called a factor. In the product ab, a is a factor, and so is b. In the product x.(a + m), x is one of the factors, and a + m, the other. Hence every coefficient may be considered a factor. In the product 3y, 3 is a factor as well as y.
41. A quantity is said to be resolved into factors, when any factors are taken, which, being multiplied together, will produce the given quantity. Thus Sao may be resolved into the two factors 3a and b, because 3a.b is 3ab. And 5amn may be resolved into the three factors 5a, and m, and n. And 48 may be resolved into the two factors 2.24, or 3.16, or 4.12, or 6.8; or into the three factors 2.3.8, or 4.6.2, &c.
42. The character / is used to show that the quantity which precedes it, is to be divided, by that which follows. Thus a/c is a divided by c: and (a + b)/(c + d) is the sum of a and b divided by the sum of c and d. But in algebra, division is more commonly expressed, by writing the divisor under the dividend, in the form of a vulgar fraction. A character prefixed to the dividing line of a fractional expression, is to be understood as referring to all the parts taken collectively; that is to the whole value of the quotient. Thus a - signifies that the quotient of b + c divided by m + n is to be subtracted from a. And denotes that the first quotient is to be multiplied into the second.
43. When four quantities are proportional, the proportion is expressed by points, in the same manner, as in the Rule of Three in arithmetic. Thus a:b = c:d signifies that a has to b, the same ratio which c has to d. And ab:cd = (a + m):(b + n), means, that ab is to cd; as the sum of a and ro, to the sum of b and n.
44. Algebraic quantities are said to be alike, when they are expressed by the same letters, and are of the same power: and unlike, when the letters are different, or when the same letter is raised to different powers. Thus ab, 3ab, -ab, and -6ab, are like quantities, because the letters are the same in each, although the signs and coefficients are different. But 3a, Sy, and Sbx, are unlike quantities, because the letters are unlike, although there is no difference in the signs and coefficients.
45. One quantity is said to be a multiple of another, when the former contains the latter a certain number of times without a remainder. Thus 10a is a multiple of 2a; and 24 is a multiple of 6.
46. One quantity is said to be a measure of another, when the former is contained in the latter, any number of times, without a remainder. Thus 3b is a measure of 15b; and 7 is a measure of 35.
47. The value of an expression, is the number or quantity, for which the expression stands. Thus the value of 3 + 4 is 7; of 3.4 is 12; of 16/8 is 2.
48. The reciprocal of a quantity, is the quotient urising from dividing a unit by that quantity. Thus the reciprocal of a is 1/a; the reciprocal of a + b is 1/(a + b); the reciprocal of 4 is 1/4.
49. The relations of quantities, which in ordinary language, are signified by words, are represented in the algebraic notation, by signs. The latter mode of expressing these relations, ought to be made so familiar to the mathematical student, that he can, at any time, substitute the one for the other. A few examples are here added, in which, words are to be converted into signs.
1. What is the algebraic expression for the following statement, in which the letters a, b,c,&c. may be supposed to represent any given quantities ?
The product of a, b, and c, divided by the difference of c and d, is equal to the sum of b and c added to 15 limes h.
Ans.abc/(c - d) = b + c + 15h.
2. The product of the difference of a and h into the sum of b, c and d, is equal to 37 times m, added to the quotient of b divided by the sum of h and b. Ans.
3. The sum of a and b, is to the quotient of b divided by c; as the product of a into c, to 12 times h. Ans.
4. The sum of a, b and c, divided by six times their product, is equal to four times their sum diminished by d. Ans.
5. The quotient of 6 divided by the sum of a and b, is equal to 7 times d, diminished by the quotient of b, divided by 36. Ans.
50. It is necessary also, to be able to reverse what is done ia the preceding examples, that is, to translate the algebraic signs into common language.
What will the following expressions become, when words are substituted for the signs ?
1. (a + b)/h = abc - 6m + a/(a + c).
Ans. The sum of a and b divided by h, is equal to the product of a, b, and c diminished by 6 times m, and increased by the quotient of a divided by the sum of a and c.
2. ab + (3h - c)/(x + y) = d.(a + b + c) - h/(6 + b).
3. (a - b):ac = (d - 4)/m :3.(h + d + y).
51. At the close of an algebraic process, it is frequently necessary to restore the numbers, for which letters had been substituted, at the beginning. In doing this, the sign of multiplication must not be omitted, as it generally is, between factors, expressed by letters. Thus, if a stands for 3, and b for 4; the product ab is not 34, but 3.4, i. e. 12.
In the following example, Let a = 3, b = 4, c = 2 and d = 6, m = 8, n = 10.
Then, 1.(a + m)/cd + (bc - n)/3d = (3 + 8)/2.6 + (4.2 - 10)/3.6.
52. An algebraic expression, in which numbers have been substituted for letters, may often be rendered much more simple, by reducing several terms to one. This cannot generally be done, while the letters remain. If a + b is used for the sum of two quantities, a cannot be united in the same term with b. But if a stands for 3, and b for 4, then a + b =3 + 4= 7. The value of an expression, consisting of many terms may thus be found, by actually performing, with the numbers, the operations of addition, subtraction, multiplica* tion, &c. indicated by the algebraic characters.
Find the value of the following expressions, in which the letters are supposed to stand for the same numbers, as in the preceding article.
1. ad/c + a + mn = 3.6/2 + 3 + 8.10 = 92.
2. (a + c).(n - m) + (m - b)/(m - d) - a.(n - m) = ?
3. (ac + 5m)/(2n + 3) + m - cb + (4d - b)(a - c)/n = ?