A L G E B R A. DEFINITIONS AND NOTATION. i. ALGEBRA LGEBRA is the science of computing by symbols, It is sometimes also called Analysis ; and is a general kind of arithmetic, or universal way of computation. 2. In this science, quantities of all kinds are represented by the letters of the alphabet. And the operations to be per Tormed with them, as addition or subtraction, &c, are denoted by certain simple characters, instead of being expressed by words at length. 3. In algebraical questions, some quantities are known or given, viz. those whose values are known: and others unknown, or are to be found out, viz. those whose values are not known. The former of these are represented by the leading letters of the alphabet, a, b, c, d, &c; and the latter, or unknown quantities, by the final letters, z, y, x, u, &c. 4. The characters used to denote the operations, are chiefly the following: + signifies addition, and is named plus. signifies subtraction, and is named minus. or . signifies multiplication, and is named into. • signifies division, and is named bye ✓ signifies the square root; the cube root; the 4th root, &c; and the nth root: ::: : signifies proportion. Thus a + b denotes that the number represented by b is to be added to that represented by a. a-b denotes, that the number represented by b is to be subtracted from that represented by a. a in b denotes the difference of a and b, when it is not known which is the greater. VOL. I. M ab, or Х of a; m ab, or a xb, or a.b, expresses the product, by multiplication, of the numbers represented by a and b. @ +- b, or , denotes, that the number represented by a is to be divided by that which is expressed by b. a:b::c:d, signifies that a is in the same proportion to b, as c is to d. x= a - b + c is an equation, expressing that x is equal to the difference of a and b, added to the quantity c. Na, or aš, denotes the square root of a; iya, or aš, the cube root of a; and yo° or aš the cube root of the square also wa, or a", is the mth root of a; and "a" or am is the nth power of the mth root of a, or it is a to the power. e’ denotes the square of a; a' the cube of a ; at the fourth power of a ; and a" the nth power of a. a+bx c, or (a+b) c, denotes the product of the compound quantity a + b multiplied by the simple quantity c. Using the bar or the parenthesis ( ) as a vinculum, to connect several simple quantities into one compound. ab-b, or expressed like a fraction, means -6 the quotient of a + b divided by a - b. vab + cd, or (ab + cd), is the square root of the compound quantity ab + cd. And av ab + cd, or c (ab + cd), } denotes the product of c into the square root of the compound quantity ab + cd. a +b-c, or (a + b + c)}; denotes the cube, or third power, of the compound quantity a + b - c. 3a denotes that the quantity a is to be taken 3 times, and 4 (a + b) is 4 times a + b. And these numbers, 3 or 4, showing how often the quantities are to be taken, or multiplied, are called Co-efficients. Also x denotes that x is multipled by ; thus x x or a + b -3 5. Like Quantities, are those which consist of the same letters, and powers. As a and 3a; or 2ab and 4ab; or 3a2bc and – 5a2bc. 6. Unlike Quantities, are those which consist of different letters, or different powers. As a and b; or 2a and a’; or 3ab and 3abc. 7. Simple / 7. Simple Quantities, are those which consist of one term only. As 3a, or 5ab, or 6abc. 8. Compound Quantities, are those which consist of two or more terms. As a + b, or 2a 3c, or a + 2b – 3c. 9. And when the compound quantity consists of two terms, it is called a Binomial, as a +b; when of three terms, it is a Trinomial, as a + 2b 3c; when of four terms, a Quadrinomial, as 2a - 36 + + 4d; and so on. Also, a Multinomial or Polynomial, consists of many terms. 10. A Residual Quantity, is a binomial having one of the terms negative. As a 26. 11. Positive or Affirmative Quantities, are those which are to be added, or have the sign +. As a or + a, or ab: for when a quantity is found without a sign, it is understood to be positive, or have the sign + prefixed. 12. Negative Quantities, are those which'are to be subtracted. As a, or - 2ab, or - Bab, 13. Like Signs, are either all positive (+), or all negative (-). 14. Unlike Signs, are when some are positive ( + ), and others negative (--). 15. The Co-efficient of any quantity, as shown above, is the number prefixed to it. As 3, in the quantity 3ab. 16. The Power of a quantity (a), is its square (a”), or cube (a?), or biquadrate (a*), &c; called also, the 2d power, power, or 4th power, &c. 17. The Index or Exponent, is the number which denotes the power or root of a quantity. So 2 is the exponent of the square or second power a’ ; and 5 is the index of the 플 cube or 3d power; and į is the index of the square root, a or va; and } is the index of the cube root, a5, or ja. 18. A Rational Quantity, is that which has no radical sign (V) or index annexed to it. As a, or 3ab. 19. An Irrational Quantity, or Surd, is that which has not an exact root, or is expressed by meas of the radical sign v. As v2, or va, or 3 d’, or abt. 20. The Reciprocal of any quantity, is that quantity inverted, or unity divided by it. So, the reciprocal of a, or 1 b ījis, and the reciprocal of is b 21. The or 3d I M2 21. The letters by which any simple quantity is expressed, may be ranged' according to any order at pleasure. So the product of a and b, may be either expressed by ab, or ba; and the product of a, b, and c, by either abc, or acb, or bac, or bca, or cab, or cba; as it matters not which quantities are placed or multiplied first. But it will be sometimes found convenient in long operations, to place the several letters according to their order in the alphabet, as abc, which order also oscurs most easily or naturally to the mind. 22. Likewise, the several members, or terms, of which a compound quantity is composed, may be disposed in any order at pleasure, without altering the value of the signification of the whole. Thus, 3a – 2ab + 4abc may also be written 3a + 4abc – 2ab, or 4abc + 3a – 2ab, or — 2ab + 3a + 4abc, &c; for all these represent the same thing, namely, the quantity which remains, when the quantity or term 2ab is subtracted from the sum of the terms or quantities 3a and 4abc. But it is most usual anòl natural, to begin with a positive term, and with the first letters of the alphabet. SOME EXAMPLES FOR PRACTICE, 1, and In finding the numeral values of various expressions, or combinations, of quantities. Supposing a = 6, and b = 5, and c 4, and d e = 0. Then 1. Will c. to 3ab ca 36 + 90 16= 110. 216 4. And of c? + 16 = 12 + 16 - 28. at 3c 13 5. And ✓ 2ac + c? or 2ac + c^) y 64 = 8. 2bc 40 6. And vit =2+ = 7. 2ac + c 8 62 36 1 35 3. And 12 7 - ac 20-N 62 of ac 8. And 62 ac + N2ac to c? 1+8 = 9. 9. And ba-ac + v 2ac+c=25 -- 244-8 = 3. 10. And ab to-d: = 183. Jl. And Oab - 1052 += 24. 12. And 24. And Va + 5-va b = 4:4936249. 25. And 3ac? + Ba – b3 = 292:497942. 26. And ta’ – 3a va'- ab = 72. ADDITION. ADDITION, in Algebra, is the connecting the quantities 'together by their proper signs, and incorporating or uniting into one term or sum, such as are similar, and can be united. As 3a + 2b - 2a = a + 2b, the sum. The rule of addition in algebra, may be divided into three cases : one, when the quantities are like, and their signs like also; a second, when the quantities are like, but their sigris unlike; and the third, when the quantities are unlike. Which are performed as follows*. CASE * The reasons on which these operations are founded, will readily appear, by a little reflection on the nature of the quantities to he |