DEFINITIONS AND NOTATION. 1. Algebra, as we have observed, is the science which teaches general methods of performing computations, by means of letters, signs, and other general characters. 2. Quantities are expressed by the small letters of the alphabet, to which the figures 1, 2, 3, &c. are sometimes joined; this will be fully explained in its proper place. 3. The sign is the mark for Addition; it is named plus, (more,) and is placed between two quantities, to shew that the quantity which follows the sign is to be added to the quantity going before it. Thus, a + b is read a plus b, and signifies that the quantity represented by b, is to be added to the quantity represented by a. 4. Quantities having the sign + prefixed, are named positive or affirmative quantities: if a quantity has no sign prefixed, it is affirmative,+ being understood; and if a positive (or affirmative) quantity stands alone, or on the left of all the others connected with it, then the sign + is usually omitted; but this sign is never omitted in any other case. 5. The sign-is the mark for Subtraction; it is named minus, (less,) and signifies that the quantity which follows the sign, is to be subtracted from the quantity going before it, or from the other quantities concerned. Thus, a-b is read a minus b, and shews that the quantity represented by b, is to be subtracted from that represented by a ; also c+a+b shews that c is to be subtracted from the sum of a and b. 6. Quantities having the sign - prefixed, are called negative quantities this sign is always prefixed to negative quantities, and must not be omitted in any case: likewise every quantity is either positive or negative, and consequently must have either +, (expressed or understood,) or belonging to it. 7. Two or more quantities are said to have like signs, when In addition to these, the capitals are frequently employed, as are the Greek letters, the signs of the ecliptic, and in general, the characters peculiar to any subject to which Algebra may be applied. the signs are either all +, or all and they are said to have unlike signs, when some are+, and others 8. The sign x is the mark for Multiplication; it is named into, and signifies that the quantities, between which it stands are to be multiplied together. Thus, a × b, is read a into b, and shews that the quantity represented by a is to be multiplied into, or by, the quantity represented by b. с 9. A point is sometimes used instead of the sign x; thus ax b may be written a.b, also 3 x x x y z may be written 3.x.y.z. 10. Both the sign and point are frequently omitted; thus a × b, or a.b, are frequently written thus, ab; also 2 x x x y is written 2xy; and when several letters are placed together like the letters of a word, their product is always understood; thus abcd denotes a x bx c x d and the same is to be understood when a number is connected with the letters; thus 3 x axbx c, is written 3abc; but when two or more numbers are connected, either the sign x must be interposed between the numbers, (not the point,) or their product must be taken; thus, 3 × 4 x x x y, must be written 3 x 4 xy, or 12xy, and not 3.4.x.y, or 34xy. 11. When the product (or multiplication) of two or more letters, is denoted by placing the letters together like a word, it is indifferent in what order they are placed; thus ab and ba are the same; also abc is the same as acb, or cba, or cab, or bea, or bac; but it is usual, for the sake of method, to place the letters in alphabetical order: likewise when a number is connected with letters by the sign x, it is indifferent in what order they stand; thus, a × 3 × b, and a × b × 3, and b × a × 3, and b x 3 x a, and 3 x a × b, and 3 × b xa, are the same; but when the sign x is not interposed, the number must always stand first, thus, 3ab, or 3ba, and never a3b, or b3a, or ab3, or ba3. 12. The sign is the mark for Division; it is named by, and signifies that the quantity standing before the sign, is to be divided by the quantity which follows it. The use of the point to denote multiplication was introduced (as Dr. Huton supposes) by M. Leibnitz. Thus, a b is read a by b, and denotes that the quantity represented by a, is to be divided by that represented by b. 12. Division is likewise denoted by placing the dividend above the divisor, with a small line between, like a fraction; thus a÷b It may be remarked, that the signs +, −, and always be long to (or govern) the quantity which follows the sign; thus in the expression a b+c... b is the quantity to be subtracted, not a; and c is the quantity to be added, not b; also in the expression a÷b... b is the divisor, and not a. But the sign of multiplication x equally affects both quantities between which it stands; thus a x b, denotes the multiplication of a by b, or of b by a, either may be considered as the multiplier. 13. The sign = is the mark for equality; it is named equal or equals, and denotes that the quantity or quantities on one side, of the sign, are equal to the quantity or quantities on the other side. Thus ab, is read a equals b, and denotes that the quantity represented by a, is equal to that represented by b; also a+b=c-d, is read a plus b equal c minus d, and shews that the sum of a and b is equal to the difference of c and d. 14. The coefficient of a quantity is the number prefixed to it. Thus in the quantities 2a, 3bd, and 4xyx, 2 is the coefficient of a, 3 is the coefficient of bd, and 4 is the coefficient of xyz; these numbers are sometimes called numeral coefficients. In quantities consisting of two or more letters, placed together like a word, any one or more of the letters may be considered as the coefficient of the rest; thus in the quantity ab... a is the coefficient of b, and b of a; in the quantity xyz x is the coefficient of yz, y of xz, and z of xy; in like manner yz is the coefficient of x, xz of y, and xy of z: these, in order to distinguish them from the numeral ones, are called literal coeffi cients. ... Every algebraic quantity has a numeral coefficient, either expressed or understood; when 1 is the coefficient, it is seldom put down, and when a quantity has no coefficient, 1 is always understood to be its coefficient; when the coefficient is greater than 1, it is always expressed. 15. A simple quantity is that which is represented by one number or letter, or by several letters standing together like a word, or by a number and letters so placed. Thus, 7 . . . ab ... 2b, and — 4xyz, are simple quantities. 16. A compound quantity is that which consists of two or more simple quantities, called its terms, connected together by the sign + or—. Thus a+b is a compound quantity, having a and b for its terms; and 2x + 3y − 4 is a compound quantity, the terms of which are 21, 3y, and — 4. It is indifferent in what order the terms of a compound quantity stand, provided every term has its proper sign prefixed; thus, a + b may be written b + a, and 2x + 3y − 4 may be written 3y+ 2x — 4, or −4 + 2x + 3y, &c. - 17. A binomial is a compound quantity consisting of two terms, as a + z . . . 2b + 4c, &c. A trinomial of three terms, as x-3y+2z, &c. A quadrinomial of four terms, as +b-3c+x. And a multinomial, or polynomial, of many terms, as a- z + 26 − c + 4, &c. - 2a 18. A residual quantity is a binomial having one of its terms negative, as a- b, or 2x — 5z, &c. 19. The powers of a quantity are the products which arise by multiplying the quantity continually into itself; they are produced and named as follows. Every quantity is considered as being the first power of itself; thus, a is the first power of a. If a quantity be multiplied once into itself, the product is called the second power, or square of that quantity; thus a xa, or aa, is the second power or square of a. If a quantity be multiplied continually twice into itself, the product is called its third power, or cube; thus a×a×a, or aaa, is the third power, or cube of a. In like manner any quantity, as a, multiplied three times into itself, produces aaaa, the biquadrate, or fourth power of a, and so on for higher powers. 20. But the powers of quantities are frequently and more conveniently represented by small figures, called indices or erponents, placed over, and a little to the right of, the quantities. Thus a2 is the same as aa, and denotes the second power or x3 is the same as xxx, denoting the third power yyyy, or y*, equally express the fourth power or square of a.. or cube of x ... |