ab, or a X b, or a.b, expresses the product, by multiplication, of the numbers represented by a and b. a b denotes, that the number represented by a is to be divided by that which is expressed by b. abcd, signifies that a is in the same proportion to b, and c is to d. x=α- bc is an equation, expressing that is equal to the difference of a and b, added to the quantity c. √ a, or a a, denotes the square root of a; a, or a3, the cube root of a; and a2 or as the cube root of the square of a; alsoa, or am, is the mth root, of a; and an or a is the 7th power of the mth root of a, or it is a to the power. m נת 772 a2 denotes the square of a; a3 the cube of a; a4 the fourth power of a; and a" the nth power of a. a + bxc, or (a + b) c, denotes the product of the compound quantity ab multiply by the simple quantity c. Using the bar, or the parenthesis () as a vinculum, to connect several simple quantities into one compound. a+b a+b÷a—b or — —, expressed like a fraction, means a-b the quotient of a + b divided by a-b. ✔ ab + cd, or (ab + ca), is the square root of the compound quantity ab + cd. And cab + cd, or c (ab + cd)3, denotes the product of c into the square root of the compound quantity abcd. a+b 3 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 denotes that is multiplied by; thus xx or 5. Like Quantities, are those which consist of the same Jetters, and powers. As a and 3a; or 2ab and 4ab; or Sa2bc and-5a2bc. 6. Unlike Quantities, are those which consist of different letters, or different powers. As a and b; or 2a and a2; or Sab2 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 26 – 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 + 26 3c; when of four terms, a Quadrinomial, as 2a. 36 + c4d; 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-2b. 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. Asa, or 2ab, or 3ab2. 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 (a2), or cube (a3), or biquadrate (a4), &c; called also, the 2d power, or 3d 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 a2; and 3 is the index of the cube or 3d power; and is the index of the square root, a ora; and is the index of the cube root, a3, or a. 18. A Rational Quantity, is that which has no radical sign (✔) or index annexed to it. As a, or 3ab. 19. An Irrational Quantity, or Surd, is that of which the value cannot be accurately expressed in numbers, as the square roots of 2, 3, 5. Surds are commonly expressed by means of the radical sign, as √2, √a, va2, or ab. 20 The Reciprocal of any quantity, is that quantity inverted, or unity divided by it. So, the reciprocal of a, or is. b 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 occurs 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 signifi cation of the whole. Thus, 3a 2ab4abc 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 and natural, to begin with a positive term, and with the first letters of the alphabet. SOME EXAMPLES FOR PRACTICE. In finding the numeral values of various expressions, or combinations, of quantities. Supposing a 6, and 6 = 5, and c = 4, and d = c = 0. Then 1, and c3 36 +90 — 16 = 110. 24. And a + b2 — √ a2 — 62 = 4·4936249. 25. And Sac2 + √ a3 -63: =292-497942. 26. And 4a2-3a √ a2 — Zab = 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, Sa+26-2a = a + 26, 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 signs unlike; and the third, when the quantities are unlike. Which are performed as follows. CASE *The reasons on which these operations are founded, will rea dily appear, by a little reflection on the nature of the quantities to CASE I When the Quantities are Like, and have Like Signs. ADD the co-efficients together, and set down the sum; after which set the common letter or letters of the like quantities, and prefix the common sign + or -. be added or collected together. For, with regard to the first example, where the quantities are 3a and 5a whatever a represents in the one term, it will represent the same thing in the other; so that 3 times any thing and 5 times the same thing, collected together, must needs make 8 times that thing. As if a denote a shilling; then 3a is 3 shillings; and 5a is 5 shillings, and their sum 8 shillings. In like manner, - 2ab and 7ab, or2 times any thing, and 7 times the same thing, make-9 times that thing. As to the second case, in which the quantities are like, but the signs unlike; the reason of its operation will easily appear, by reflecting, that addition means only the uniting of quantities together by means of the arithmetical operations denoted by their signs and , or of addition and subtraction; which being of contrary or opposite natures, the one co-efficient must be subtracted from the other, to obtain the incorporated or united mass. As to the third case, where the quantities are unlike, it is plain that such quantities cannot be united into one, or otherwise added, than by means of their signs; thus, for example, if a be supposed to represent a crown, and b a shiling; then the sum of a and b can be neither 2a nor 2b, that is neither 2 crowns nor 2 shillings, but only 1 crown plus 1 shilling, that is a + b. In this rule, the word addition is not very properly used; being much too limited to express the operation here performed. The business of this operation is to incorporate into one mass, or algebraic expression, different algebraic quantities, as far as an actual incorporation or union is possible; and to retain the algebraic marks for doing it, in cases where the former is not possible. When we have several quantities, some affirmative and some nagative; and the relation of these quantities can in the whole or in part be discovered; such incorporation of two or more quantities into one, is plainly effected by the foregoing rules. It may seem a paradox, that what is called addition in algebra, should sometimes mean addition, and sometimes subtraction. But the paradox wholly arises from the scantiness of the name given to the algebraic process; from employing an old term in a new and more enlarged sense. Instead of addition, call it incorporation, or union, or striking a balance, or any name to which a more extensive idea may be annexed, than that which is usually implied by the word addition; and the paradox vanishes. Thus, |