ALGEBRA. DEFINITIONS. ART. 1. ALGEBRA is a branch of mathematics in which calculations are performed by means of letters which denote numbers or quantities, and signs which indicate operations to be performed on them. 2. The first letters of the alphabet, as a, b, c, &c., are used to denote known quantities, and the latter letters, as X, Y, Z, &c., to denote unknown ones. 3. The sign + (named plus), indicates that the quantities between which it stands are to be added together; thus a+b denotes the sum of the quantities a and b. 4. The sign — (named minus), indicates that the number or quantity placed after it is to be subtracted from that placed before it; thus a—6 denotes the remainder left by taking the quantity b from a. 5. The sign X (named multiplied into), indicates that the quantities between which it stands are to be multiplied the one by the other; thus a xc denotes that a is to be taken as often as there are units in c, or that c is to be taken as often as there are units in d. This symbol is however seldom used, as a.c, or simply ac written as the letters of a word, indicates the same thing. 6. The sign ; (named divided by), indicates that the quantity before it is to be divided by that placed after it; thus a-c denotes that a is to be divided by c. bol is also seldom used, as division is more commonly denoted by placing the dividend above a line as the numerator of a fraction, and the divisor below it as its denominator; thus is the same as a=6. 7. The sign = (read equal, or is equal to), indicates that the quantities before it are equal in value to those after it; thus 4x3+7=9x2+1, for each is equal to 19. 8. The quantities before and after the sign = are together called an equation; that portion which stands before being called the first side of the equation, and the portion after it the second. This sym the sign B 9. The symbol y denotes that the number over which it is placed is to have its square root extracted; thus N 16 indicates the square root of 16, which is 4, and the Na denotes the square root of a, that is a number that, being multiplied into itself, would produce a. 10. In the same manner, the cube root of a number as a is denoted by 3ā, the fourth root, by Va, and so on. 11. A number placed before a letter or combination of letters is called a coefficient; thus 3a denotes three times a, and 3 is called the coefficient of a. The first letters of the alphabet are frequently called the coefficients of the latter letters; thus, in the expression 3cx, 3c is called the coefficient of x. 12. When the same letter enters several times as a multiplier into an expression, instead of repeating the letter it is only written once, and a figure written after it to indicate the number of times it enters as a multiplier; thus a”, a3, a4, &c., denote respectively the second, third, and fourth powers of a, and the small figures, 2, 3, 4, &c., placed after the letters, are called the exponents or indices of the letters. 13. Fractional exponents are also used to indicate roots; thus, instead of Nx, xt, is written, for 3*, x3, for x, a#, and so on to any extent; fractional exponents, where the numerator is not one, are also used; thus x3, x7, &c., the former of which denotes that x is to be raised to the second power, and then the third root of this power extracted, and the latter denotes that x is to be raised to the fifth power, and then the square root of this power extracted; or generally the numerator of the fractional exponent denotes a power to which the quantity is to be raised, and the denominator indicates the root of this power which is to be extracted. 14. When = as it is frequently written thus, a:0:: c:d, and read, a is to b as c is to d, and the four quantities are said to constitute a proportion or analogy; the terms a and d are called extremes, and b and c means. 15. The symbol .. is used instead of the words therefore or consequently, which occur very frequently in mathematical reasoning; and the symbol : instead of because. 16. Like quantities are such as are expressed by means of the same letters, and the same powers of these letters, and unlike quantities are expressions which contain different letters or different powers of the same letters; thus 3a%c3 and 7a2c5 are like quantities, whilst 5bx’y and 96ʻxy? are unlike. 17. A simple quantity consists of one term, as 4cx; a compound quantity consists of two or more terms connected by the signs + or — ; thus 16a+c+ab and 13c-x3—4cd are compound quantities. 18. A vinculum, bar ---, or parenthesis ( ), is used to collect several quantities into one; thus a + xd or (a + x)d denotes that the sum of a and x is to be multiplied into d; also v4acB or (4ac-—62) indicates the square root of the remainder left by subtracting the square of b from four times the product of a multiplied into c. 19. The reciprocal of a quantity is the quotient arising from dividing unity by that quantity; thus - is the reciprocal of a, and can also be written a-'; in the same manner the reciprocals of a», «5, z", are un or a ', or x x", or ", where n may represent any number either whole or fractional, and is used as a general symbol for any exponent. 20. Find the numerical values of the following expressions, when a=8, b=4, c=3, d=2, e=15, f=0. 1. ac +(1-de-bcd 30 2. a(bc+e)-d(be—c) 3. (a-1)(6–1)(0-1)(d+c) = 210 4. apce tb-ce+b 35 5. aʻ(be—ce)-f 2ad +6 = 960 6. a}(26+2+4)3 +33(etc—a); 14 In the following equations, the first and second sides will always give the same numerical value, if the same value be given to the letters on each side: verify this. 1 = 102 = 7. =x + xy +ya. XY 8. (a + x)(a—x)=a?-?. 9. (a+b4c)(a-6+c)=a2_6-c? +2bc. 10. 54y4=(x,y)(x3 +x+y + xy2 +y5). ADDITION 21. Is commonly divided into three cases:-1st, When the quantities are like, and have like signs; 2d, When the quantities are like, but have unlike signs; 3d, When the quantities are not all like, and have unlike signs. CASE I. RULE.—Add the coefficients together, and to their sum annex the literal part. EXAMPLES. 1st. 2d. ac a 3d. . 4th. За 54 x2 + y2 2a'bcxy? 7a Бас 374 +32 5a2bcxy? 7a-bcxya 19 (x2 +ya)* 5a-bcxy? 4ac 4. x2 + y2 2abcxy? 5a Зас 3 (2c2 + y2 )] 8a%bcxy? Sum, 26a -29ac 35 (x2+y2) 29a%bcxy Ex. 1. Find the sum of ax} + {ax +3ax} +44ax: + Baxt. Ans. 9ļax. 2. Find the sum of 2amx3 + 4amx3 +7<amx3 +4 amx3. Ans. 17amoc. 3. Find the sum of 47 x2 +44 +7 7x++y4 + W x +y4 +37x2 + y4 +19 7x2 + y4 +107x++y4. Ans. 44**+y'. 4. Find the sum of 12(74—2)$+(24—2)+7(x*—z)} +4x4. x+)+. Ans. 41|(x4—z)). 22. Case II. When the quantities are like, but have unlike signs. Rule.-Add the coefficients of the plus quantities into one sum, and those of the minus quantities into another, their difference is the coefficient of the sum, and is plus if the sum of the plus coefficients be the greater, and minus if it be the less. |