Exercises for the Slate or Black-board. 1. What is the cube root of 262,144? Ans. 6 is the greatest root in the second period, the unit's figure is 4: 64, therefore, is the root. 2. What is the cube root of 389,017? Ans. 7 being the greatest cube in 389, and 3 the complement of 7, the cube root is 73. 3. Find the cube roots of the following perfect powers by a glance : 54,872 ; 884,736 ; 185,193 ; 474,552 ; 5832 ; 15,625 ; 59,319. Prove by involution by inspection. P The extraction of the square root by this method requires more attention ; but for that very reason is more useful as a mental exercise. The roots of perfect squares which do not exceed 10,000 may be ascertained thus. By an examination of the squares of the digits, it will be perceived that every perfect square ending in 5 has 5 for the unit's figure of its root; and that the squares of 1, 2, 3, 4, and of their complements, 9, 8, 7, 6, have for unit's figure 1, 4, 9, 6, respectively. Thus, having determined the ten’s figure of the root by a glance at the second period, we know that if the unit's figure of the square be Exercises for the Slate and Black-board. 1. What is the square root of 5184 ? Ans. The greatest square in the second period being that of 7, and the remainder small compared with 7, the unit must be 2 rather than 8. The whole root, of course, is 72. 2. What is the square root of 1296 ? Ans. The greatest square in 12 is that of 3, and the remainder being great in proportion to 3, must be 6 rather than 4. The whole root, then, is 36. 3. What is the square root of 2025? Ans The greatest square in 20 is 4, and as the unit's figure in the square is 5, that of the root must be 5 also. The whole root, then, is 45. 4. Find the square roots of 1024 ; 3136 ; 784 ; 4225; 2116 ; 3249; 6561 ; 2401. Prove by involution by inspection, SYNOPSIS, OR, RECAPITULATION OF PRINCIPLES DEVELOPED IN THE PRECEDING PAGES. I. THERE are only two operations in arithmetic, increase and decrease. A number may be increased by one or more additions. It may be dimin ished by one or more subtractions. Such is the whole sum and substance of arithmetic. Both of these can be performed by numeration. All other processes are mere abbreviations of this foundation of arithmetic by the omission of steps become superfluous by repeated practice. II. In adding or subtracting integers, the numbers must be of one denomination. Four thousand added to three hundred, can only make four thousand three hundred. The 4 and the 3 do not make 7 of any denomination. For a similar reason, 3 hundred cannot be taken from 7 thousand, so as to leave just 4 of any denomination whatever. The latter operation can only be performed by changing the denomination of one of the thousands into hundreds, and then subtracting the 3 hundred from the 10 hundred, leaving six thousand and seven hundred. Precisely the same remarks apply to fractional quantities. In order that fractions may be united into one number, their denomination must be the same. Three fourths and four fifths, or three pecks and four bushels, can be united or subtracted, only after some change of denomination which renders them similar. No such restriction, however, is necessary in multiplication or division. The product of 4 thousand or 4 hundred by 2 or by 2 ty can be found just as readily as if the denominations were the same ; and the quotient of 4 million by 2 is as easily found as the quotient of 4 by the same number. The reason is obvious. A product is not one factor increased by another. It is the amount of the one taken as many times as there are units in the other ; and a quotient merely points out how many times one number is contained in another. And this is not less so with fractions than with integers. For { can be taken } times, and we can find how many times for can be found in f, or ; and the same remarks apply equally to bushels, yards, &c., as to 3ds or 7ths. But, after all, the difference is only in appearance. Multiplication is, in fact, simply an addition of identical numbers, while division is the subtraction of numbers equally identical. It is only the manner in which the operations are performed which gives them the appearance of different denominations. Bringing the numbers to the same denomination is one of the steps that become superfluous, by the use of multiplication and division in place of addition and subtraction. See p. 91. III. The object of all arithmetical operations is to produce a balance, or, in technical language, to form an equation. Thus, both in integers and fractional quantities, In addition, we seek to find a number=the sum of the given numbers. In all the changes of fractional quantities we seek to place a fraction, without altering its value, in a more convenient or simple and intelligible form. In proportion, the sole object is to change the complete ratio into the same denomination with that of the imperfect ratio, so that the term wanting in the latter may appear. In other words, to find a fraction of the denomination of the imperfect ratio=to the complete ratio. IV. When figures are written horizontally, or side by side, whether they be integers or decimal fractions, every figure is tenfold greater than the same figure immediately on its right, and tenfold less than the same figure immediately on its left. See p. 114. V. Every figure becomes tenfold greater by being removed one rank or place towards the left, tenfold less by being removed one rank or place towards the right, p. 115. VI. Ten units of any one place make one unit of the next place to the left ; and one unit of any one place makes ten units of any one place to the right, p. 117. VII. When there is a separatrix, the unit's place is immediately on its left ; when there is none, the right hand figure occupies the unit's place, p. 119. When a separatrix is used, any number of figures or ciphers can be placed to the right of a number, without changing the value of any of the figures in that number, p. 118, 1. 27. VIII. The cipher is superfluous unless it occupies the place of units, or intervenes between a significant figure and the place of units, p. 122. IX. If equal numbers be added to unequal numbers, their difference remains unchanged, p. 139. X. 1. If the difference between two numbers be added to the smaller, it becomes equal to the greater ; 2. If taken from the greater, it becomes equal to the smaller, p. 140. XI. The first significant figure on the right of an arithmetical complement is always greater by 1 than if it stood one or more ranks to the left, p. 143, i. 6. XII. The difference between two numbers may be obtained by adding the complement of the smaller to the larger, and diminishing this sum by 1 of the next higher rank of figures than is contained in the smaller, p. 144. XIII. In multiplication, the number of decimal fractional places in the product is always equal to the number in both factors, and, consequently, as the dividend is the product of the divisor and quotient, there will always be as many fractional places in the dividend as in the divisor and quotient; and, conversely, as many in the divisor and quotient as in the dividend. In the same manner the number of ciphers at the right of a product is always equal to the number at the right of both factors, p. 154. XIV. An integer may have any number of decimal places annexed to it, by affixing a separatrix to show the place of units, and then adding as many ciphers as may be required. Thus, if 48 is to be divided by '4, by changing the dividend to 48*0, the quotient is the whole number 120, p. 118, 1. 27. XV. 1. The SQUARE of any number of tens and units=the squares of the tens and of the units taken separately, plus twice the product of the tens and units. 2. The CUBE of any number of tens and units=the cubes of the tens and of the units taken separately, plus three times the square of the tens multiplied by the units, and three times the square of the units multiplied by the tens, p. 166. 5. By multiplying the denominator, the fraction is XVI. In common fractions, 1. The numerator expresses the number, the denominator its denomi. nation, or value, p. 67, 8. 2. The denominator expresses the number of parts into which a unit is divided, the numerator the number of those parts which the fraction contains.- 1b. 3. The numerator is the dividend, the denominator the divisor, and both terms taken together the quotient. — Ib. is dividing dividedp. 80. s divided. { multiplied, p. 81. 6. By multiplying dividing both, the value of the fraction is unchanged, p. 82. 7. By removing the denominator, the fraction is multiplied by a number equal to the denominator. 8. Any number, whether fractional or integral, may be represented in an infinite variety of forms, all differing in character, yet all exactly equal, p. 82. 9. Any number whatever may be expressed by a common fraction, whose numerator (or whose denominator) shall consist of any specified number, whether whole or fractional, p. 211. 11. Any denominator may be used to express any number whatever. XVII. An integer or a decimal fraction is changed to a common fraction by expressing its secondary name under it in figures ; a common fraction is changed to a decimal fraction or to an integer by performing the division indicated, p. 193. XVIII. Circulating decimals with a simple repetend are changed to common fractions by using the repetend as numerator, and as many 9s as there are figures in the repetend as denominator. XIX. Circulating decimals, with compound repetends, may be changed to common fractions, by changing separately the repetends and the figures that precede them to common fractions, and then adding them together, p. 198, 1. 32. XX. In changing complex fractions of three terms to simple fractions, when the upper term is an integer, the upper and lower terms are factors of the numerator, and the other term is the denominator ; but, when the lower term is the integer, the upper term is the numerator, and the other two are factors of the denominator, p. 203. XXI. The form of a fraction can always be conveniently changed by multiplication ; but by division it frequently becomes more intricate, p. 92, 1. 10, and p. 200, 1. 14. XXII. The same quotient will result, whether division is performed by a composite number, or by its factors separately, p. 86, 1. 24. XXIII. Addition or subtraction of common fractions is performed by exactly the same process as addition or subtraction of integers, p. 91, XXIV. Signs for discovering Factors of Numbers, 1. Every even number is necessarily divisible by 2. For, in dividing the tens, the remainder must either be 0 or 1. If it be nothing, the last number to be divided will be 0, 2, 4, 6, or 8; if it be 1, the last number must be 10, 12, 14, 16, or 18; and all of these are divisible by 2, p. 85. 2. If the two right hand figures of a number are divisible by 4, or by 25, the whole number is divisible by 4 or by 25. For both those numbers will divide one hundred, and consequently any number of hundreds without remainder. the number of tens be even, we need not look further than the units to decide if the number be divisible by 4, p. 86. 3. If the three right hand figures of a number are divisible by 8, or by 125, the whole number is so divisible. For as one thousand is divisible by both those numbers, so, of course, is any number of thousands. If the number of hundreds be even, we need not look further than the tens to decide whether the whole number be divisible by 8. Therefore, when the hundreds are odd, we have only to render them even by taking one hundred with the tens and units. Thus 953752 is divisible by 8 because 152 is divisible by 8, every even number of hundreds being already known to be so divisible, p. 86. 4. Every number terminated on the right by 0, or by 5, is divisihle by 5, because ten or any number of tens is divisible by 5, p. 86. 5. If the sum of the significant figures of any number, added horizontally, be divisible by 3, or by 9, the whole number is divisible by 3 or by 9. Thus, the number of 6273 is divisible by 9, because, since 1000 consists of 999 and 1, it will leave a remainder of 1 when divided by 9, and consequently 6000 will leave a remainder of 6. For a like reason 200 will leave a remainder of 2, 70 a remainder of 7, and 3 units a remainder of 3. Now, as the remainders in this division are its significant figures, this must evidently be so in any number whatever. If, therefore, the sum of the significant figures is divisible by 9, the whole number is so divisible. The same demonstration will answer for 3. Therefore, every number, the sum of whose significant figures is divisible by 9 or by 3, is itself divisible by 9 or by 3. It is obvious, also, that every number divisible by 9 is also divisible by 3, though the converse by no means always follows, namely, that every number divi ble by 3 is also divisible by 9, p. 85. 6. Every even number divisible by 3 is also divisible by 6. For 2 is a factor in every even number, and 2X3=6; and it is evident that the products of factors of any number are also factors in that number. For the same reason, every even number divisible by 9 is also divisible by 18, p. 86. 7. Every number divisible by 3 and by 4 is also divisible by 12 ; every number divisible by 3 and by 5 is also divisible by 15 ; every number divisible by 3 and by 8 is divisible by 24 ; and every number divisible by 9 and by 25 is divisible by 225, p. 86. 8. Table of the foregoing signs for the discovery of factors : 2 Even numbers. |