To find the Roots of Equations by Approximation, To find the Roots of Exponential Equations, Of the Indeterminate Analysis, Of the Summation and Interpolation of Series, The Rule of Three by Logarithms, ALGEBRA. ALGEBRA is the science which treats of a general method of performing calculations, and resolving mathematical problems, by means of the letters of the alphabet. Its leading rules are the same as those of arithmetic; and the operations to be performed are denoted by the following characters : + plus, or more, the sign of addition; signifying that the quantities between which it is placed are to be added together. Thus, a + b shows that the number, or quantity, represented by b, is to be added to that represented by a; and is read a plus b. minus, or less, the sign of substraction; signifying that the latter of the two quantities between which it is placed is to be taken from the former. Thus ab shows that the quantity represented by b is to be taken from that represented by a: and is read a minus b. Also, ab represents the difference of the two quantities a and b, when it is not known which of them is the greater. X into, the sign of multiplication; signifying that the quantities between which it is placed are to be multiplied together. Thus, a × b shows that the quantity represented by a is to be multipled by that represented by b; and is read a into b. The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word. ; Thus, a × b, a. b, and ab, all signify the product of a and b; also, 3 × a, or 3a, is the product of 3 and a; and is read 3 times a. by, the sign of division; signifying that the former of the two quantities between which it is placed is to be divided by the latter. Thus, a÷b, shows that the quantity represented by a is to be divided by that represented by b; and is read a by b. or a divided by b. Division is also frequently denoted by placing one of the two quantities over the other, in the form of a fraction b both signify the quotient of b divided Thus, ba and ab a+c a by a; and a + c. equal to, the sign of equality; signifying that the quantities between which it is placed are equal to each other. signifies that a - b is to be divided by Thus, x = a + b shows that the quantity denoted by x is equal to the sum of the quantities a and b; and is read x equal to a plus b. Any two algebraic expressions are said to be identical, when they are of the same value, for all the values of the letters of which they are composed. * Thus (x + a) × (x − a) x2-a2, whatever numeral values may be given to the quantities represented by x and a. > greater than, the sign of majority; signifying that the former of the two quantities between which it is placed is greater than the latter. Thus a > b shows that the quantity represented by a is greater than that represented by b; and is read a greater than b. < less than, the sign of minority; signifying that the former of the two quantities between which it is placed is less than the latter. Thus, a < b shows that the quantity represented by a is less than that represented by b; and is read a less than b. : as, or to, and so is, the signs of an equality of ratios; signifying that the quantities between which they are placed are proportional. Thus, ab:: : d denotes that a has the same ratio to b that c has to d, or that a, b, c, d, are proportionals; and is read, as a is to b so is c to d, or, a is to b as c is to d. ✓ the radical sign, signifying that the quantity before which is placed is to have some root of it extracted. 22 * Woodhouse, in his Principles of Analytical Calculation, says that x2-a2 is not generally = (x—a). (x+a): for instance, the particular case of x = a is to be excluded; the proof essentially demanding this circumstance, to wit, that x -a be a quantity, or that x be greater than = x 1 an identical equation; and shows that x is indeterminate, or that any number whatever may be substituted for it. See Euler's Algebra, page 289, Vol. I.—Ed. a. Euler calls x I: Thus, va is the square root.of a;a is the cube root of a; and a is the fourth root of a; &c. 4 The roots of quantities are also represented by figures placed at the righthand corner of them, in the form of a fraction. 1 2 Thus, a is the square root of a; a3 is the cube root of a; and an is the nth root of a, or a root denoted by any number n. 1 In like manner, a2 is the square of a ; a3 is the cube of a; and am is the mth power of a, or any power denoted by the number m. is the sign of infinity, signifying that the quantity standing before it is of an unlimited value, or greater than any quantity that can be assigned, The coefficient of a quantity is the number or letter which is prefixed to it. 3 Thus, in the quantities 36, b, 3 and 2 are the coefficients of b; and a is the coefficient of x in the quantity ax. A quantity without any coefficient prefixed to it is supposed to have 1 or unity; and when a quantity has no sign before it, is always understood. Thus, a is the same as + a, or + la; and a is the same as la. A term is any part or member of a compound quantity, which is separated from the rest by the signs + or Thus a and b are the terms of a+b; and 3a, +5cd, are the terms of 3a 2b + 5cd. In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed. d b a Thus, a and b are the terms of ab, or of and a, b, c, d, 26, and are the terms of the proportion a:b::c: d. A factor is one of the terms, or multipliers which form the product of two or more quantities. Thus, a and b are the factors of ab; also, 2, a, and b2, are the factors of 2ab2; and a ≈ and b x are the factors of the product (ax) × (bx). A composite number, or quantity, is that which is produced by the multiplication of two or more terms or factors. Thus, 6 is a composite number, formed of the factors 2 and 3, or 2 × 3; and 3abc is a composite quantity, the factors of which are 3, a, b, c. Like quantities, are those which consist of the same letters or combinations of letters; as a and 3a, or 5ab and 7ab, or 2a2b and 9a2b. Unlike quantities, are those which consist of different letters, or combinations of letters; as a and b, or 3a, and a2, or 5ab2 and 7a2b. Given quantities, are such as have known values, and are generally represented by some of the first letters of the alphabet; as a, b, c, d, &c. Unknown quantities, are such as have no fixed values, and are usually represented by some of the final letters of the alphabet; as x, Y, Z. Simple quantities are those which consist of one term only; as 3a, 5ab, — 8a2b, &c. Compound quantities, are those which consist of several terms; as 2a + b, or 3a -2c, or a + 2b 3c, &c. Positive, or affirmative quantities, are those which are to be added; as a, or + a, or + 3ab, &c. Negative quantities are those which are to be subtracted: a, or 3ab, or 7ab2, &c. Like signs, are such as are all positive, or all negative; as + and +, or and as Unlike signs, are when some are positive and others negative; as and or and +. " A monomial, is a quantity consisting of one term only; as a, 2b, — 3a2b, &c. A binomial, is a quantity consisting of two terms, as a + b, or a b; the latter of which is, also, sometimes called a residual quantity. A trinomial, is a quantity consisting of three terms, as a2b3c; a quadrinomial of four, as a-2b+3c-d, and a polynomial, or multinomial, is that which has many terms. The power of a quantity, is its square, cube, biquadrate, &c.; called also its second, third, fourth power, &c.; as a3, a3, aa, &c. The index, or exponent of a quantity, is the number which denotes its power or root. Thus, 1 is the index of a-1, 2 is the index of a2, and Voic 10 of aor √ a. When a quantity appears without any index, or exponent, it is always understood to have unity, or 1. Thus, a is the same as a1, and 2x is the same as 2x1; the 1, in such cases, being usually omitted. |