3. Required the square root of 63. 4. Required the cube root of fa3b. 5. Required the 4th root of 16a2. 6. Required to find the —mth root of a". 7. Required the square root of a2 · 6a √b + 9b. Ans. +66. Ans. 2√ a. II. To extract the square root of a binomial quadratic surd as of a ± √b. (1.) The product of two quadratic surds will be a surd, except when one of them is some rational multiple (integer or fractional) of the other. y = √mx, Let y = mx then and xy mx2 = = x/m. where, except ✅m be rational, the result is irrational. Hence, if m be rational, y is a rational multiple of √x; if not, not. Whence the proposition is true. (2). One quadratic surd cannot be the sum or difference either of a rational quantity and a quadratic surd, or two quadratic surds. For, first, assume √ = x ± √y; then squaring, z = x2 + 2 x √y + y, Whence, if we suppose √2=x+ √y, we shall have a surd equal to a rational quantity, which is contrary to the definition of a surd. Again, secondly, assume √z = √ z = x + 2√x + y, or √xy √y: then squaring, we have = ± 11 (2 xy), or again, a surd equivalent to a rational quantity, which is contrary to definition. This demonstration may be objected to as incomplete, inasmuch as √x and ✔✅y may be the one a rational multiple of the other, in which case √xy will be rational. Then, however, the expression would take the form √2 = (1 + m) √x; and multiplying both by z, we have z = (1 ± m) √xz, and the equation cannot hold good, except √✅z be a rational multiple of √x, or the converse. In this case we have then simply √z = p √x = q √y, and the equation would take the form √2 = (+)/z, or the surd factor ✅z is simply a multi plier of every term of the equation, and should be rejected, leaving the expression entirely rational. It has now been proved that the equations √2 = x + √y and √z = √x ± √y cannot have a real existence; and, therefore, that whenever they occur they are the result of incompatible conditions amongst the data of the inquiry. They constitute in fact one of the many forms of the imaginary symbol. It is to be understood that x, y, z, are to be perfectly general in their nature, and not restricted to special numbers. It readily follows from this, that in the equation a ± √b = x ± √y, we must have a = a, and √y = √b. Assume a x + a: then x + a ±√b = x±√y, and hence, ± √y = a ± √b, which has been proved impossible, except a = 0: and then a = a, and ± √y = ± √b. (3.) To extract the square root of a binomial surd of the form a ± √b. Assume a±√b = = u± √v, or squaring we have a ± √b= u + v ± 2 √uv; and hence, u + v = a, and ± 2√uv = ± √b. In resolving these equations, we shall have successively, 2 It is quite obvious that, except √a2 - b he a rational quantity, the formula, on the right side of the equation here obtained, is far less convenient than that on the left. If this criterion be fulfilled, the solution will be more simple, and the method will be of advantage; but if not, it is better to calculate a ± √b, and then extract its square root. As the criterion is always easy to apply, it is desirable to do so in the outset, and then be guided by its result in the choice of the method to be employed in the actual numerical valuation. 4. Divide a+b- c+2 √ab by a + √b+ √c, and by √a + √b−√c. √xy by √x + √y; and a25 + a2b3 + a1·sz3 + ab + 6. Write the following expressions with fractional and (where possible) de ax cimal indices : \√/a, 3√ (a + 2)2, "√ (aa — ba)", Jun; m by' express, by means of radicals, the following quantities: -a; {(a—x)−1}}; 7. Ascertain whether any of the following expressions are rational : { (a + b + x)' .5 3; ± √ a2 + b2 + x2 + 2ab + 2ax ‡ 2bx ; √ (a + bx)a xy ; 3√ (a + x)2b2; · x2)m (a + x) 8. Reduce a2. ; and { (3 (a2 — x2) {a2 — 4 ax + x2 } } } . 4 ax + x2 } } . 5b (a + x) (ax)3 cs) · x2 and 1√a1 + a1 to others having a common index }; and (a1 — x-1)-25 and (a — x) 25 to others having the common index — 5. I 11. Express the mth root of the nth root of a" in as many ways as possible; (1 − x) √/1 + 2x + x2; of 28 + 5 √12; and of √/32 — √/24. Scholium. AMONGST the examples already given in this work, the symbol + √−1 as a co-efficient has appeared. This is called the imaginary symbol, and expressions into which it enters are called imaginary quantities. This symbol is one which indicates an operation that cannot be performed in real numbers, positive or negative. For since (+1)2 and (—1)2 are alike unity, -1 cannot be the result of squaring either of them; whilst the direction to extract the square root of -1 implies that -1 has been produced by squaring some quantity, and which quantity it is required to assign. All, then, that can be done is to prefix the radical symbol, giving the general form ± √−1. In all problems where this symbol appears in the result, there has been some incongruity in the conditions of the question; or in other words, the alleged conditions would not co-exist. The calculus by means of these is precisely similar to those laid down for quadratic surds, and no remark seems necessary, except to caution the student to use due care in the signs of his reduced quantities. The laws, however, are precisely the same as already laid down for products and quotients. ARITHMETICAL PROGRESSION, OR PROGRESSION OF DIFFERENCE. [THOUGH in accordance with the original arrangement of the Course, the subject of progressions is retained in this place, it is very desirable that the study of it should be deferred till the simple and quadratic equations are thoroughly understood. The investigations at least, as they depend on the solution of simple equations, will be unintelligible to the student who has read no farther than the preceding pages.] An Arithmetical Progression is a series of quantities which either increase or decrease by the same common difference. b Thus, 1, 3, 5, 7, 9, 11, ... and a, a + b, a ± 2b, a ± 3b, a ± 4b, . . . are series in arithmetical progression, whose common differences are and respectively. When the quantity b is affected by sign +, the progression is said to be increasing; and when by the sign it is said to be a decreasing progression. These signs are therefore the indications of the kind of progression. The most useful part of arithmetical progression has been given in the arithmetic. The same may be exhibited algebraically, thus : Let a denote the least term, z the greatest term, d the common difference, n the number of the terms, and s or s, the sum of n terms; then the principal relations are expressed by these equations, viz. 1. The first term of a series of quantities in arithmetical progression is 1, their difference 2, and the number of terms 21: what is the sum ? + 2; and n = 21: and hence, by formula (5), we have d 1+ 41 2 X 21 441. 2. A decreasing arithmetical series has its first term 199, its difference and its number of terms 67: what is its sum? 3, 3. To find the sum of 100 terms of the natural numbers 1, 2, 3, 4, 5, 6, &c.; and likewise of 1, 2.... Ans. 5050, and 505. 4. Required the sum of 99 terms of the negative odd numbers — 5, — 7, — 9.... Ans. 5. The first term of an arithmetical series is 10, the common difference — }, and the number of terms 21: required the sum of the series. Ans. 140. 6. One hundred stones being placed on the ground, in a straight line, at the distance of two yards from each other: how far will a person travel, who shall bring them one by one to a basket, which is placed 2 yards from the first stone? Ans. 11 miles and 840 yards. 7. The first term is a2 2ax+x2, the last is a2 + 2ax + x2, and the number * For, writing the given series first in a direct line and then in an inverted order, we have {a + (n − 1) d} + {a + (n−2)d} + {a + (n−3)d} + and adding up the columns vertically, we have double the sum of the + {a+(n−d)d} + {a series equal to ...... } {2a + (n − 1)d} + {2a + (n − 1)d} + {2a + (n − 1)d} + + {2a + (n − 1) d} which is equal to n{2a + (n − 1)d}, there being ʼn such terms; and hence s1 = : {a + {(n − 1)d}n. This is the formula marked (5) above. That marked (1) is obvious from the formation of the successive terms: the (2) is obtained from it by transposition. Also from the method of proceeding in the proof of (5) that marked (3) is evident; and substituting the value of a in (5) the result (4) is obtained. From those expressions which involve values of n and d, formulæ for finding these may be derived by the common operations of algebra: but it is unnecessary to annex the work here. x2: what is the sum of the series, and the common difference + 4ax of its terms? Ans. Suma1- x1, com. dif. = 8. The first term is a, and the last term is nine times the first, and the number of terms one-fifth of the sum of the first and last terms: what is the 10. The first term is 16.5, the last is 6.5, and the sum is 100. Find the number of terms and the common difference. 11. The tenth term of an arithmetical series is 17.5, and the fiftieth is 1587. What are the separate sums of the first twenty, and of the last thirty terms? Find also the common difference; and the 11th, 21st, 31st, and 41st terms. and find the 7th term from each end of each of these series. 13. The common difference is 001; the number of terms is one million, and the greatest term is 0. What is the least? Find also the sum of the 100th, 200th, 300th, .... terms of which the series is composed. 14. The first term is 1, the common difference is successively taken 1, 2, 3, ... write ten terms of the first six series, and express the sum of each series to n terms. .... 15. Find the difference between the sum of n terms of the odd numbers 1, 3, 5, .... and n terms of the series of even numbers 0, 2, 4, 16. If the first term be a, and the common difference be 2a (1+a+a2+...aTM—3), show that the sum of the series of a terms is equal to a". And apply this to the square and cube as values of m. d, and the number of nth term? Also, in +n- 1 17. Let the first term be-a, the common difference terms — n*, what is the expression for the value of the numbers where a = + 5, d =2, and n = 10. Ans. za± 2 d. 18. A triangular battalion † consists of thirty ranks, in which the first rank is formed of one man only, the second of 3, the third of 5, and so on: what is the strength of such a triangular battalion ? Ans. 900 men. 19. A detachment having 12 successive days to march, with orders to advance * As n denotes the number of terms to be taken in the direction indicated by the sign of the common difference, so also -n denotes the term from which we must have started to obtain -a (the difference still being d) as the nth term of the series. By triangular battalion, is to be understood, a body of troops ranged in the form of a triangle, in which the ranks exceed each other by an equal number of men: if the first rank consist of one man only, and the difference between the ranks be also 1, then its form is that of an equilateral triangle; and when the difference between the ranks is more than one, its form may then be an isosceles or scalene triangle. The practice of forming troops in this order, which is now laid aside, was formerly held in greater esteem than forming them in a solid square, as admitting of a greater front, especially when the troops were to make simply a stand on all sides. |