SECTION III. DEFINITIONS AND NOTATION. [Continued from Section I.] 17. THE last letters of the alphabet, x, y, z, &c., are used in algebraic processes to represent unknown quantities, and the first letters, a, b, c, &c., are often used to represent known quantities. NUMERICAL QUANTITIES are those expressed by figures, as 4, 6, 9. LITERAL QUANTITIES are those expressed by letters, as a, x, y. MIXED QUANTITIES are those expressed by both figures and letters, as 3a, 4x. 18. The sign plus, +, is called the positive or affirmative sign, and the quantity before which it stands a positive or affirmative quantity. If no sign stands before a quantity, is always understood. 19. The sign minus,, is called the negative sign, and the quantity before which it stands, a negative quantity. 20. Sometimes both and are prefixed to a quantity, and the sign and quantity are both said to be ambiguous; thus, 8 ± 3 = 11 or 5, and a ± b = a + b, or ab, according to circumstances. 21. The words plus and minus, positive and negative, and the signs + and —, have a merely relative signification; thus, the navigator and the surveyor always represent their northward and eastward progress by the sign +, and their southward and westward progress by the sign, though, in the nature of things, there is nothing to prevent representing northings and eastings by and southings and westings by +. So if a man's prop erty is considered positive, his gains should also be considered positive, while his debts and his losses should be considered negative; thus, suppose that I have a farm worth $5000 and other property worth $3000 and that I owe $1000, then the net value of my estate is $5000 + $3000 $1000 $7000. Again, suppose my farm is worth $5000 and my other property $3000, while I owe $12000, then my net estate is worth $5000 + $3000 $12000 — $4000, i. e. I am worth $4000, or, in other words, I owe $4000 more than I can pay. From this last illustration we see that the sign may be placed before a quantity standing alone, and it then merely signifies that the quantity is negative, without determining what it is to be subtracted from. 7y, 22. The TERMS of an algebraic expression are the quantities which are separated from each other by the signs + or -; thus, in the equation 4 a - b = 3 x + c the first member consists of the two terms 4 a and and the second of the three terms 3x, c, and b, 23. A COEFFICIENT is a number or letter prefixed to a quantity to show how many times that quantity is to be taken; thus, in the expression 4x, which equals x + x +x+x, the 4 is the coefficient of x; so in 3 ab, which equals ab+ab + ab, 3 is the coefficient of a b; in 4 ab, 4a may be considered the coefficient of b, or 4b the coefficient of a, or a the coefficient of 4b. Coefficients may be numerical or literal or mixed; thus, in 4 ab, 4 is the numerical coefficient of ab, a is the literal coefficient of 4b, 4 a is the mixed coefficient of b. If no numerical coefficient is expressed, a unit is understood; thus, x is the same as 1x, bc as 1bc. 24. An INDEX or EXPONENT is a number or letter placed after and a little above a quantity to show how many times that quantity is to be taken as a factor; thus, in the ex pression b3, which equals bb b, the 3 is the index or exponent of the power to which b is to be raised, and it indicates that b is to be used as a factor 3 times. An exponent, like a coefficient, may be numerical, literal, or mixed; thus, x3, x, x31, &c. b If no exponent is written, a unit is understood; thus Coefficients and Exponents must be carefully distinguished from each other. A Coefficient shows the number of times a quantity is taken to make up a given sum; an Exponent shows how many times a quantity is taken as a factor to make up a given product; thus 4 x = x + x + x + x, and x = x x x x x x x. 25. The product obtained by taking a quantity as a factor a given number of times is called a power, and the exponent shows the number of times the quantity is taken. 26. A Rooт of any quantity is a quantity which, taken as a factor a given number of times, will produce the given quantity. A Roor is indicated by the radical sign, , or by a fractional exponent. When the radical sign,, is used, the index of the root is written at the top of the sign, though the index denoting the second or square root is generally omitted; thus, x, or x, means the second root of x; Every quantity is considered to be both the first power and the first root of itself. 27. The RECIPROCAL of a quantity is a unit divided by that quantity. Thus, the reciprocal of 5 is, and of x, 28. A MONOMIAL is a single term; as a, or 3x, or 5 bxy. 29. A POLYNOMIAL is a number of terms connected with each other by the signs plus or minus; as x + y, or 3 a + 4x 7aby. 30. A BINOMIAL is a polynomial of two terms; as 3x + 3y, or x y. 31. A RESIDUAL is a binomial in which the two terms are connected by the minus sign, as x y. 32. SIMILAR TERMS are those which have the same powers of the same letters, as x and 3x, or 5 ax3 and 2 ах3. But x and x2, or 5 a and 5 b, are dissimilar. 33. The DEGREE of a term is denoted by the sum of. the exponents of all the literal factors. Thus, 2 a is of the first degree; 3 a2 and 4 ab are of the second degree; and 6 a3 x1 is of the seventh degree. 34. HOMOGENEOUS TERMS are those of the same degree. Thus, 4 a2 x, 3 abc, x2y, are homogeneous with each other. 35. To find the numerical value of an algebraic expression when the literal quantities are known, we must substitute the given values for the letters, and perform the operations indicated by the signs. The numerical value of 7 a = 2, and c = 5 is 7 × 4 +25=37. b EXAMPLES. b+c2 when a = 4, 24+ 52 = 28 16 Find the numerical values of the following expressions, b: = 13, c = 4, d = 15, m = 5, and n = 7. when a = 2, 1. a + b 2. a2 + 3bc Ans. 41. Ans. 40. Ans. 219. n -- C. B 20. ab 10 (d—m) + 14c. 36. Write in algebraic form : 1. The sum of a and b minus the difference of m and n. (m>n.) 2. Four times the square root of the sum of a, b, and C. 3. Six times the product of the sum and difference of c and d. (ed.) 4. Five times the cube root of the sum of a, m, and n. 5. The sum of m and n divided by their difference. 6. The fourth power of the difference between a and m. |