25. A term, is any part or member of a compound quantity, which is separated from the rest by the signs and ; thus, a and b are the terms of a+b; and 3a, —2b, and +5ad, are the terms of the compound quantity 3a-2b+5ad. In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed; thus, a and b are the terms of ab, or of ; and a, b, c, d, are the terms of the proportion a: b::c: d. α b 26. A measure, or divisor, of any quantity, is that which is contained in it some exact number of times; thus, 4 is a 35a measure of 12, and 7 is a measure of 35a, because -5a. 17 27. A prime number, is that which has no exact divisor, except itself, or unity; 2, 3, 5, 7, 11, &c. and the intervening numbers; 4, 6, 8, &c. are composite numbers. (Art. 11.) 28. Commensurable numbers, or quantities, are such as have a common measure; thus, 6 and 8, 8ab, and 4ab, are commensurable quantities; the common divisors being 2 and 4; also, 4ax2 and 5ax are commensurable, the common divisor being ax. 29. Also, two or more numbers are said to be prime to each other, when they have no common measure or divisor, except unity; as 3 and 5, 7 and 9, 11 and 13, &c. 30. A multiple of any quantity, is that which is some exact number of times that quantity; thus, 12 is a multiple of 4; and 15a is a multiple of 3a, because 15a 3a 5. 31. The reciprocal of a quantity is that quantity inverted or unity divided by it. Thus, the reciprocal of a, or ofis α a-b a+b b reciprocal of is and the reciprocal of is b α a+b a 1 the a 32. The reciprocal of the powers and roots of quantities, is frequently written with a negative index or exponent; 1 thus, the reciprocal of a2, may be written a2; the re ciprocal of (a+x)3= 1 a2 (a+x)31 may be written (a+x); but this method of notation requires some farther explanation, which will be given in a subsequent part of the work. 33. A function of one or more quantities, is an expression into which those quantities enter in any manner whatever, the same combination of letters, or that differ only in their numeral coefficients; thus, 5a and 7a; 4ax and 9ax; +2ac and 9ac; -5ca; &c., are called like quantities; and unlike quantities are such as consist of different letters, or of different combination of letters; thus, 4a, 3b, 7ax, 5ay2, &c. are unlike quantities. 20. Algebraic quantities have also different denominations, according to the sign+, or Positive, or affirmative quantities, are those that are additive, or such as have the sign + prefixed to them; as, +α, +6ab, or 9ax. 21. Negative quantities are those that are subtractive, or such as have the sign - prefixed to them; as, —î, —3a2, -4ab, &c. A negative quantity is of an opposite nature to a positive one, with respect to addition and subtraction: the condition of its determination being such, that it must be subtracted when a positive quantity would be added, and the re verse. 22. Also quantities have different denominations, according to the number of terms (connected by the signs or ) of which they consist; thus a, 3b, -4ad, &c., quantities consisting of one term, are called simple quantities, or monomials; a+x, a quantity consisting of two terms, a binomial; a-x is sometimes called a residual quantity. A trinomial is a quantity consisting of three terms; as, a+2x-3y; a quadrinomial of four; as, a-b+3x-4y; and a polynomial, or multinomial, consists of an indefinite number of terms. Quantities consisting of more than one term may be called compound quantities. 23. Quantities the signs of which are all positive or all negative, are said to have like signs; thus, +3a, +4x, +5ab, have like signs; also, -4a, -3b,,-4ac; When some are positive, and others negative, they have unlike signs; thus, the quantities +3a and -5ab have unlike signs; also, the quantities-3ax, +3a2x: and the quantities-b, tb. 24. If the quotients of two pairs of numbers are equal, the numbers are proportional, and the first is to the second, as the third to the fourth; and any quantities, expressed by such numbers, are also proportional; thus, if; then a is to bas c to d. The abbreviation of the proportion; a:b::c: d; and it is sometimes written a:b=c:d; if a=8, b=4, 8 12 c=12, and d=6; then, =2, and 8:4:; 12: 6. x=8, and y=12; then, to find its value, we have ab+ax by=5X45X8—4×12 =20+40-48 =60-48-12. 3ax+2y Ex. 3. What is the value of ,where a 4, x=5, y a+b ? = = 10, and b=6? Here 3ax+2y=3×4×5+2×10=60+20=80, and a+ b=4+6=10; a+b 10 Ex. 4. What is the value of a2+2ab-c+d, when a=6, b=5, c=4, and d-1? Ans. 93. Ex. 5. What is the value of ab+ce- bd, when a=8, b=7, c=6, d=5, and e-1? Ans 27. Ex. 6. In the expression ax+by let a = 5, b=3, x=7, b+x and y=5; what is its numerical value?. Ans. 5. Ex. 7. In the expression bx-a2-c' let a 3, b=5, c=2, x=6; What is its numerical value? Ans. 7. Ex. 8. What is the value of ax(a+b)-2abc, where a=6, b=5, and c=4? Ans. 156. Ex. 9. There is a certain algebraic expression consisting of three terms connected together by the sign plus; the first term of it arises from multiplying three times the square of a by the quantity b; the second is the product of a, b and c; and the third is two thirds of the product of a and b. Required the expression in algebraic writing, and its numerical value, where a=4, b=3, and c=2? Ans. 176. DEFINITIONS. 38. A proposition, is some truth advanced, which is to be demonstrated, or proved; or something. proposed to be done or performed; and is either a problem or theorem. 39. A problem, is a proposition or question, stated, in order to the investigation of some unknown truth; and which requires the truth of the discovery to be demonstrated. 40. A theorem, is a proposition, wherein something is advanced or asserted, the truth of which is proposed to be demonstrated or proved. 41. A corollary, or consectary, is a truth derived from some proposition already demonstrated, without the aid of any other proposition. 42. A lemma, signifies a proposition previously laid down, in order to render more easy the demonstration of some theorem, or the solution of some problem that is to follow. 43. A scholium, is a note, or remark, occasionally made on some preceding proposition, either to show how it might be otherwise effected; or to point out its application and use. 44. An axiom, is a self-evident truth, or proposition universally assented to, or which requires no formal proof. 45. As axioms are the first principles upon which all mathematical demonstrations are founded, I will point out those that are necessary to be observed in the study of Algebra, as there will be frequent occasion to advert to them. AXIOMS. 46. When no difference can be shown or imagined between two quantities, they are equal. 47. Quantities equal to the same quantity, are equal to each other. 48. If to equal quantities equal quantities be added, the wholes will be equal. Thus, if a = b, then a+c=b+c; if a-b-c, then adding b, a-b+b=c+b, or a=c+b. 49. If from equal quantities equal quantities be subtracted, the remainders will be equal. If a=b, then, a—2—b−2; if b+c=a+c, then b=a. 50. If equal quantities be multiplied by equal numbers or quantities, the products will be equal. b Thus, if a=b, 3a=3b; if a= 3a=b; if a=b, ca=cb; 3' and if a=b, aXa=bb or a2=b2. 51. If equal quantities be divided by equal numbers or quantities, the quotients will be equal. Ба 106 Thus, if 5a=10b, = or a=2b; if ca = cb, 5 5 a2 ba or a= =b. or a=b; and if a2= ba, then a a ca cb C = Scholium. Articles (49), (50), (51), might have been deduced from Art. (48); but they are all easily admitted as axioms. 52. If the same quantity be added to and subtracted from another, the value of the latter will not be altered. Thus, if a=c, then a+b=c+b, and a+b−b=c+b―b, or a=c. |