a Thus, a Xo shows that the quantity d by a is to be multiplied by that represented 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 Xb, a . b, and ab, all signify the product of a and b : also, 3Xa, 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. Thus, bąd and both signify the quotient of $ d? b , vided by a; and signifies that a - b is to be divided ato by atc. equal to, the sign of equality ; signifying that the quantities between which it is placed are equal to each other. Thus, x=a + b shows that the quantity denoted by a is jual to the sum of the quantities aʼand b; and is read qual to a plus b. = identical to, or the sign of equivalence ; signifying that the expressions between which it is placed are of the same value, for all values of the letters of which they are composed. Thus, (x+a) X (2-a) =x_a>, whatever numeral values may be given to the quantities represented by a a amb 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 sigos of an equality of ratios ; signifying that the quantities between which they are placed are proportional. Thus, a : 6 :: c :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 it is placed is to have some root of it extracted. Thus, va is the square root of a; Va is the cube root of a ; and V a is the fourth root of a ; &c. The roots of quantities, are also represented by figures placed at the right hand corner of them, in the form of a fraction. Thus, ais the square root of a ; aš is the cube root } of a ; and a ñ is the nth root of a, or a root denoted by any number n. In like manner, ais the square of a; a3 is the cube of a; and am is the mtb power of a, or any power den noted by the number m. is the sign of infinity, signifying that the quantity 1 3 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. Thus, in the quantities 36, - 36, 3 and are the -6 coefficients of b; and a is the coefficient of u 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 + mort la ; and — 'a is the la. A term is any part or member of a compound quantity, which is separated from the rest by the signs + same as ::c: d. Thus, a and b are the terms of a +b; and 3a, 26, and + 5cd, are the terms of 3a – 26 + 5cd. 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, o, d, are the terms of the proportion a : 6 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 62, are the factors of 2ab2 ; and a - x and b -X are the factormf the product (a —c) (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 2x3; 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 2a8b and 9a2b. Or Unlike quantities, are those which consist of umerent letters, or combinations of letters ; as a and b, or 30 and az, 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, 2. Simple quantities, are those which consist of one term only ; as 3a, 5ab, – 892b, &c. Compound quantities, are those which consist of several terms ; as 2a *-b, or 30 - $c, or a +2b - 3c, &c. Positive, or affirmative quantities, are those which are to be added ; as a, or ta, or +3ab, &c. Negative quantities, are those which are to be sub. tracted ; as - Q, or – 3ab, or - 7ab2, &c. Like signs, are such as are all positive, or all negative; 3 + and t, or - and 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, – 322b, &c. A binomial, is a quantity consisting of two terms; as atb, or amb; the latter of which is, also, sometimes called a residual quantity. A trionomial, is a quantity consisting of three terms, as a +26 - 3c ; a quadrinomial of four, as a- 2b + 34 d; and a polynomial, or multinomial, is that which has a many terms. The power of a quantity, is its square, cube, biqua. drate, &c. ; called also its second, third, fourth power, &c. ; as a?, as, ao, &c. The index, or exponent of a quantity, is the number which denotes its power or root. : of as, Thus, - 1 is the index of a?, 2 is the index of a? , and { of afora. When a quantity appears without any index, or exponent, it is always understood to have unity, or 1. Thus, a is the same as a', and 2x is the same as 22" ; the 1, in such cases, being usually omitted. A rational quantity, is that which can be expressed in finite térms, or without any radical sign, or fractional index ; as a, ora, or 5a &c. An irrational quantity, or surd, is that which has no exact root, or which can only be expressed by means of the radical sign, or a fractional index ; as v2 or 21, yas or aš, &c. A square or cube number, &c. is that which has an exact square or cube root, &c. Thus, 4 and a2 are square numbers ; and 64 and a3 are cube numbers? &c. A measure of any quantity, is that by which it can Di divided without leaving a remainder. Thus, 3 is a measure of 6, 7a is a measure of 35a, and 9ab of 27 0262. Commensurable quantities, are such as can be each divided by the same quantity, without leaving a remainder. Thus, 6 and 8, 2 /2 and 3 V2, 5a2b and 7ab2, commensurable quantities ; the common divisors being 2, 72, and ab. Incommensurable quantities, are such as have no mon measure, or divisor, except unity. Thus, 15 and 16, 2 and 3, and a + b and a2 + 62, are incommensurable quantities. A multiple of any quantity, is that which is some exact number of times that quantity. Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 20a3b2 of 5ab. are com |