extract the indicated root. In actual numerical practice, however, it is always best to extract the root first, when the number admits of an exact root: but when it does not admit of an exact one, it is better to raise the indicated power first, and then extract the indicated root. The reason is, that the number of figures necessary is always less than would be required in taking the contrary course. Respecting the verbal enunciation of these expressions, it should be remarked that at is generally read the third root of a, or the one-third power of a. 3 a is read the two-thirds power of a, or the cube root of a2, or the square of the cube root of a, and so on with other expressions. It ought also to be understood that the fractional indices may be, and often are, converted into decimals; as for at may be written as, for a may be 12. Another method of indicating the extraction of roots is by the symbol ✓ prefixed (being only a modification of the form of the old manuscript r, the initial letter of the word radix); and the order of the root expressed by a small number written over it, and lying a little to the left: as a, b, (a + x), Va 62; which signify respectively the same thing as a, b, (a + x), (a2-b2), or in words, the square (or second) root of a, the cube (or third) root of b, the fourth root of a + x, and the nth root of a2 The distinction of rational and irrational, in respect to quantities whose roots can or cannot be respectively extracted, is a very convenient one. An irrational quantity is often called a surd. Rational quantities are sometimes put under an irrational form, to facilitate their combination with irrational quantities into one expression. 13. Quantities receive different designations which are found useful in algebraical enunciations, according to peculiar circumstances. The following are the principal ones : A simple quantity * is that which consists of a single term, or of several factors only, each of which is a single term. As 3a, or 5ab, or 6a2b3c7, or 3a-5. It is often called a monome or a monomial quantity. A compound quantity is composed of the aggregation of two or more simple quantities connected together by addition or subtraction: as a + b, 2a - 3c, or a + 2b - 3с. Of compound quantities, that which is composed of two terms, as x + y, xy + ab, or x2 - y2t, or 3x + 4aby is called a binomial quantity; when three terms, x2 + ax + b, it is called a trinomial quantity; when four or more terms, a polynomial quantity, or simply a polynomial. When the binomial, trinomial, or polynomial expressions are so related that their terms counterbalance or mutually destroy one another in the aggregate, the * The term expression is often applied to any combination of algebraical quantities. + The term residual was formerly applied to that form of the binomial expression when they were connected by the sign, as a -b, a2-y2. The distinction is now, however, fallen greatly into disuse. total expression is put equal to zero, and the expression in this form is called a binomial equation, a trinomial equation, or a polynomial equation. Sometimes the word equation is understood, and the terms binome or binomial, trinome or trinomial, polynome or polynomial, are used instead of the compound phrase. 14. When any general numerical relation is exhibited in the form of an equation, the expression is called a formula. 15. When the numerator and denominator of any quantity are interchanged, the resulting expression is called the reciprocal of the former. If the quantity be in the integer form, it may be put in a fractional form, and its denominator is understood to be unity or 1. In that case the result or reciprocal is 1 divided by the given quantity. Thus reciprocal of cd. a+b a a-b 1 and is the cd 16. There is also a distinction to be made between the notation of factors when writing algebraically and numerically. It was explained in the third definition, that when algebraical symbols of factors were written, they were generally put in juxtaposition, without any mark between them. In arithmetic, the figures placed in juxtaposition have each a relative value derived from the particular position they occupy, which is generally called the local value, and are thereby rendered units, tens, hundreds, &c. or tenths, hundredths, &c. To represent the compound number, then, when the component figures are supplied in literal notation, each one must have for its coefficient such a power of 10 as will raise or lower it to its proper locality in the decimal scale. Thus to represent 5305-2906 without employing the artifice of local value, we must write it thus: 5 × 103 + 3 × 102 + 0 × 101 + 5 x 10° +2×10−1 + 9×10-2 + 0×10-3 + 6 x. 10-4. Or, to designate a number whose digits to the left of the decimal point were x, y, z, and the right of it were t and u, we should have 102x + 10'y +10°z + 10-1t + 10-2น. 17. An algebraical expression is said to be ranged according to the powers (or dimensions) of some quantity, when the term containing the highest power of that letter is placed first, the term containing the next power next to it in order, and so on to the lowest. Sometimes also it is so ranged as to begin with the lowest and ascend to the higher, in order. In expressions containing a finite number of terms the former plan is most commonly adopted; but in expressions where the terms run out to infinity (as it will appear in the course of the work is often the case), the latter is necessarily employed. Thus, in the expression x3 + 3x2- 9x + 10-8x-1+3x-2, the arrangement is according to the powers of æ, and so also is 3x-2 - 8x-1+10-9x + 3x2+x3, so arranged. The former in a descending series of powers, the latter in an ascending series of powers. In x2 + 2xy + y2 they are ranged according to the descending powers of æ, and in y2 + 2xy + x2 in descending powers of y. Of the infinite series of ascending EXERCISES IN NOTATION AND ITS INTERPRETATION. 109 III. Exercises in Notation and its interpretation. I. Write in words the signification of the expressions 2 - 10x = 119; x{a-x+3}$ }} y √x2 + y2 + √x2 - y2; a2 + a*; and x - y = 0. II. Write in the common numerical forms the quantities 105 × 3 + 10o × 7 + 104 x 2; and form an expression whose digits are decimals, beginning at the third place from the unit's place, are z, o, y, o, x, o, o, t. III. How are the following expressions to be interpreted verbally? (a); (a); 3√(a - x) = (a - 5x); 3a - a - b = (a + b); 6 (ax - by) = 6ах 778 66= - 4a + 156; 6by; */ a + b; 4z"; (42); 4"z"; (abc), a3bbccc, (ab)2abc". And point out any which are identical in value though different in their forms. IV. Put into algebraical symbols the following statements : 1. There is a certain number at present unknown, to the square root of which if we add its square we shall obtain 18: and another whose half exceeds its third part by three-tenths of an unit. 2. Of three unknown numbers, the sum of the first two is equal to 5, the difference of the second and third is 1, and the sum of all three is 9. 3. Add three given numbers together, and indicate the cube of the square root of their sum, multiplied by the product of the three numbers themselves. 4. Express that three times the square of a certain unknown quantity is equal to half the product last mentioned. 5. The sum of two unknown numbers is equal to the cube of the square root of their product; and their difference is equal to 10. 6. The number 6 is subtracted from 5, and the square root of the remainder is taken from the cube root of their sum; how is this to be expressed algebraically, and how in the condensed form of arithmetical notation? 7. Express the cube of the cube root of a known quantity, and the cube root of the cube of the same quantity. 8. Express that the sum of a geometrical series whose first term, ratio, and number of terms at present unknown is to be computed:-by both methods of notation. 9. The three leading terms of a proportion are always supposed given : express that the fourth (yet unknown) is equal to the product of the second and third divided by the first. And express that the square roots of four given quantities are reciprocally proportional: and also that the reciprocals of the squares of the fifth roots of four other quantities are directly proportional. 10. Of three given quantities express the sum of each two diminished by the remaining one: and the product of the three resulting quantities. 11. Denote that the difference of the cubes of two unknown quantities divided by the difference of the quantities themselves, is equal to the sum of the squares added to the product of those two quantities. 12. Express the theorem that if the sum of any number of quantities (first supposed given and then supposed unknown) be multiplied by another given quantity, the product is equal to the sum of the products made by multiplying each of the first-named (whether given or unknown) quantities by the one lastnamed. 13. The following formulæ are to be explained in words : (a + b - c)"; (2 - 4 + 6) = 2; √x2+2xy + y2 = +(x+y) In what do these last expressions differ? 14. How is the following theorem to be expressed: -the product of two known powers of an unknown quantity is equal to that power of the same quantity which is equal to the sum of the two known powers? Express the same when all the quantities are known, when all are unknown, and when two unknown powers of a known quantity are substituted in the theorem. 15. Suppose that in the conditions of some particular example that was proposed, the known indices were found to be 3 and 2, and from some additional conditions it was otherwise found that the base a was 001, what would be the value of the expression? And what if the given powers were 3 and 2? 16. Indicate the extraction of the 10th root of the 3d root of 62 1000 and of 17. Express the product of the sum of the square roots of three given quantities into the negative square of an unknown quantity being equal to the product of all the quantities mentioned. 18. Interpret the expression a : : : : ; and admitting the rules given for the "rule of three" to be true, how is the solution to be expressed in letters? and, also, if a = 10, b = 008, and c = 27000, assign the value of x. 19. A certain number is unknown: but it is known that if three times its defect from 10 be divided by 2, the quotient will be equal to one-third of double the square of the number multiplied by its square root. Express this statement in appropriate symbols. 20. What kind of symbol is? What kind is ( )? Point out and distinguish the symbols of operation from those of quantity in the expression {3/(a + b)2 - (a - b)2} x 4 cdx ÷ (a + b) (a - b) = x2. 21. If the figures which compose a number were s, r, q... d, c, b, a, reckoning a that to the right hand, and suppose the decimal point fall between e and d: How is that number to be represented algebraically ? V. Range - 3 10x + 15x2 - 8x4 - 3x + 4x-1 - 6x-2, according to powers of æ, both ascending and descending; and 4x3y - Зху3 + 9x2y2 + 4x4 -4y+, according to ascending and descending powers first of æ and then of y. Arrange xyz + xy2z2 + x^z + x*z - 3x2y2z + 4x3yz + 9xyz5 - 10x5y + 3z*x + zty, according to powers of z, and the several multipliers in terms of x and y collected in vincula, and each of these arranged respectively in ascending terms of y, and in ascending terms of x. The intelligent teacher can select a few, (such as may suit his purpose, and the defects he observes in his pupils) from the questions in the application of simple and quadratic equations: by which means the nature of algebraical notation will be completely illustrated. EXAMPLES FOR PRACTICE. In finding the numeral values of various expressions, or combinations, of quantities. Supposing a = 6, b = 5, c = 4, d = 1, and e = 0. Then will 1. a2 + 3ab c2 = 36 + 90 - 16 110. 2. And 2a3-3a2b + c = 432 - 540 + 64 -44. 3. And a2 x (a + b) - 2abc 36 x 11 240156. +16=12+ 16 = 28. 9. And b2- ac + √2ac + c2 = √25 - 24 + 8 = 3. 10. And arb + c − d = 183; and 9ab - 1062 + c = 24. a2b C C a+b b 11. Likewise xd=45; and x = 13; and a2b xe = 0. 12; a2b C 12. Show which of the two quantities is used to signify the numbers + 10, and + 100, successively. 2x + 1 3 a+b C a-b d is the greater, when a 001, -1, — 10, — 100; + 001, + 1, 13. Also keeping the old values of a, b, c, d, e, show that (b - c) x (d-e) = 1; (a + b) - (cd) = 8; and (a + b) - c - d = 6. 14. Also that ac × d3 = 144 ; acd - d = 23; are + be + d = 1; and 292 497942; and 4a2 - 3a √a2 - ab = 72. 16. Suppose a = 6 × 104; b = 5 × 103; c = 4 × 102; d = 8 × 10'; and e=1: what will be the values of the expressions in examples 11, 12, 13, and 14 ? 17. If we have in any algebraical problem reason to know that a = 0, b = 6, c = '1, d = '001, and e = c, show what the values of the expressions 1, 2, 3, 4, 5, and 6, would then be. 18. If a = 6; b = 5; c = 4; d = 1; e = 0; show what the results of the first eleven expressions would be. Likewise when a, b, c, d, e, were half these last-named values, and also four-fifths of them; three times as much, and ten thousand times as much; and write down these results adapted to each of these cases. 19. Find the values of the expressions in 11, 12, 13, when a = 1; b = 3; c = ; d= ; and e = . 20. Let a = 10, what are the values of 3 × a5, 2 × a2, 6 x a, 3 xa, and 9 xa? Also assign their sum. 1 |