.3765 .1236 .2529 their difference will be .2529. Again, let .7562 be taken from .82; by annexing ciphers to the greater and arranging the numbers thus : .8200. .7562 .0638 82 8200 we shall find the difference to be .0638: it must be observed, that the value of a decimal is not increased nor decreased by annexing ciphers to it; for a fraction does not alter its value by annexing ciphers to its numerator and denominator, thus; 100, and so on. This is also evident from the decimal notation, which is similar to that of whole numbers; that is, the value of the decimal .82 is 8 tenths and 2 hundredths, the value of the decimal .820 is also the same, being 8 tenths 2 hundredths and 0 thousandths; and so on. Examples in Subtraction of Decimals. Ex. 1. Required the difference between 57.49 and 5.768. Ans. 51.722. Ex. 2. Required the difference between .0076 and 00075. Ans. .00685. Ex. 3. Required the difference between 3.468 and 1.2591. Ans. 2.2089. Ex. 4. Required the difference between 3.1416 and .5226. Ans. 2.6180. From the multiplication of fractions, (or even the decimal notation,) it appears evident, that the multiplication of decimals is performed as in whole numbers, but if there be not as many decimals in the product as there are in both factors, ciphers must be prefixed to supply the deficiency. For instance, the product of .06 x .004 is equal to .00024; since .06 is equal, and .004 is equal o‰0; hence, Too-108800, which, expressed according to the decimal notation, is equal to .00024. Examples in Multiplication of Decimals. Ans. 8.59375. Ans. 36.37575. Ex. 1. Multiply 3.125 by 2.75. Ex. 2. Multiply 79.25 by .459. Ex. 3. Multiply .135272 by .00425. Ans. .000574906. Ex. 4. Multiply .004735 by .0375. Ans. .0001775625. The Division of Decimals is performed in the same manner as that of whole numbers, but the dividend must contain as many decimal figures as the divisor, if not, ciphers must be annexed; and the decimals in the quotient must be always equal to the excess of the decimal figures in the dividend above those in the divisor, if not, ciphers must be prefixed. 3 TO 300 For instance, when the denominators of any two fractions are the same, their quotients are found by dividing their numerators: thus, is equal to 25÷5; that is, 5: hence .025.005 is equal to 5; that is, a whole number. Again, 100 is equal to T, which is equal to 300-75, hence to the decimal .3, the dividend, two ciphers must be added, in order to have as many decimal places as the divisor .004, before the division can be performed. 1000' 375 1000 375 1000 It likewise follows, that is equal to 3+ 50%, which is equal to 375; this by reduction is equal to 7, which may be written .75; hence, .375 ÷ .5 = .75 ; that is, the decimal figure in the quotient is equal to the excess of the decimal figures in the dividend above those in the divisor. Examples in Division of Decimals. Ex. 1. Divide .1342 by 67.1. Ex. 2. Divide 1.7144 by 1.5. Ans. .002. Ans. 1.142955. Ex. 3. Divide 24880 by 360. Ans. 69.111, &c. or 694. Ex. 4. Divide 172.8 by .144. Ex. 5. Divide .88 by 88. Ans. 1200. Ans. 100. Ex. 6. When the diameter of a circle is 1 the circumference is 3.14159 nearly; what is the diameter of the earth, allowing its circumference to be 24880 miles? Ans. 7919.53666 miles, nearly. Extraction of the Square Root. The square of the sum of two numbers is equal to the squares of the numbers with twice their product. Thus, the square of 24 is equal to the squares of 20 and 4 with twice the product of 20 and 4; that is, to 400+2×20×4+16= 576. Here in extracting the second root of 576, we separate it into two parts, 500 and 76. Thus, 500 contains 400, the square of 20, with the remainder 100; the first part of the root is therefore 20, and the remainder 100+76, or 176. Now, according to the principle above mentioned, this remainder must be twice the product of 20, and the part of the roots still to be found, together with the square of that part. Now, dividing 176 by 40, the double of 20, we find for quotient 4; then this part being added to 40, the sum is 44, which being multiplied by 4, the product 176, is evidently twice the product of 20 and 4, together with the square of 4. The operation may, in every case, be illustrated in the same manner. Hence the following rule for extracting the square root of any number. Commencing at the unit figure, cut off periods of two figures each, till all the figures are exhausted, the first. figure of the square root will be the square root of the first period, or of the greatest square contained in it, if it be not a square itself. Subtract the square of this figure from the first period; to the remainder annex the next period for a dividend; and, for part of a divisor, double the part of the root already obtained. Try how often this part of the divisor is contained in the dividend wanting the last figure, and annex the figure thus found to the parts of the root and of the divisor already determined. Then multiply and subtract as in division; to the remainder bring down the next period; and, adding to the divisor the figure of the root last found, proceed as before. For instance, the square root of 106929, is found thus: Square. Root. 9 62) 169 124 647)4529 If any thing remain, after continuing the process till all the figures in the given number have been used, proceed ́in the same manner to find decimals, adding, to find each figure, two ciphers. If the root of a fraction be required, let the fraction be reduced to a decimal, and then proceed as in the extraction of the roots of whole numbers. Examples in extracting the Square Root. Ex. 1. Required the square root of 24 or 2.25. Answer, 1.5. Ex. 2. Required the square root of 152399025. Ans. 12345. Ex. 3. Required the square root of 5499025. Ans. 2345. Ex. 4. Required the square root of 36372961. Ex. 5. Required the square root of 10.4976. Ex. 6. Required the square root of 9980.01. Ex. 7. Required the square root of 2. Ans. 6031. Ans. 3.24. Ans. 99.9. Ans. 1.414213, nearly. Extraction of the Third or Cube Root. The cube or third power of the sum of two numbers is equal to the cubes of the numbers increased by 300 times the square of the first number multiplied by the second, and also increased by 30 times the first multiplied by the square of the second, thus: Multiplied 20+42 Multiplied. 20×20+4×20 +4x20+16 5 20×20+2×4×20+16=2d power 20x20x20+2x4x20x20+20×16 4×20×20+2×20×16+64 Third power 8000+3x4x20×20+3×20×16+64 or 8000+300x4x4+30×2×16+64. Hence, this rule for extracting the third or cube root of any given number:-Commencing at the unit figure, cut off periods of three figures each till all the figures of the given number are exhausted. Then find the greatest cube number contained in the first period, and place the cube root of it in the quotient. Subtract its cube from the first period and bring down the next three figures; divide the number thus brought down by 300 times the square of the first figure of the root, and it will give the second figure; add 300 times the square of the first figure, 30 times the product of the first and second figures, and the square of the second figure together, for a divisor; then multiply this divisor by the second figure, and subtract the result from the dividend, and then bring down the next period, and so proceed till all the periods are brought down. For instance, in finding the cube root of 48228544, the operation will stand thus: 48'228'544(364 root. 27 3276)21228 19656 393136)1572544 1572544 Divided by 300×32=2700 Divided by 362×300=288800 If any thing remains, add three ciphers, and proceed as before; but for every three ciphers that are added, one decimal figure must be cut off in the root. And if the cube root of a fraction or a mixed number be required, reduce the fraction to a decimal, and proceed as in whole numbers: the decimal part however must consist of periods of three figures each, if not, ciphers must be added. |