In order to understand the process of Decimal Division, it is needful to recollect the method of dividing and multiplying, by ciphers and a separatrix. If we wish to multiply a number by a sum composed of 1 with ciphers added to it, we add as many ciphers to the multiplicand, as there are ciphers in the multiplier. Thus if we wish to multiply 64 by 10, we do it by adding one cipher, 640. If we are to multiply by 100, we add two ciphers thus, 6400, &c. EXAMPLES. Multiply 3 by 100 Multiply 46 by 100 1000 2 6 100000 19 6 If we wish to multiply a decimal by any number com. posed of 1 with ciphers annexed, we can do it by removing aratrix as many orders to the right, as there are ci. phers in the multiplier. Thus if ,2694 is to be multiplied by 10, we do it thus ; 2,694. If it is to be multiplied by 100, we do it thus ; 26,94. If it is to be multiplied by 1000 we do it thus ; 269,4. But to multiply by a million, we must add ciphers also, in order to be able to move the separatrix as far as required, thus; 269400, EXAMPLES. Multiply 2,64 by 10 | Multiply 6,4 by 10000 "s36,9468 100 1,643 10 3,2 * 1000 3,2 6 1000000 The same method can be employed in dividing deci. mals, by any number composed of 1 and ciphers annexed. The rule is this. Remove the separatrix as many orders to the left, as there are ciphers in the divisor. Thus if we wish to divide 23,4 by 10, we do it thus ; 2,34. If we wish to divide it by 100 we do it thus, ,234. But if we wish to divide it by a thousand it is necessary to pre. fix a cipher thus, ,0234. If we divide it by 10,000 we do it thus, ,00234. It is needful to understand, that a mixed decimal can be changed to an improper decimal fraction. For example, if we change 3,20 to an improper decimal fraction, it becomes 320 hundredths (188), which is an improper fraction, because its numerator is larger than the denominator. But we cannot express the denominator of 320 hun. dredths, by a separatrix in the usual manner, for the rule requires the separatrix to stand, so that there will be as many figures at the right of it, as there are ciphers in the denominator. If then we attempt to write 320 hundredths in this way, it will stand thus, 3,20, which is then a mixed decimal and must be read three units and 20 hundredths. If it is writ. ten thus, 136, it is then a vulgar and not a decimal fraction. What is the rale for dividing decimals by any number composed of 1 and ciphers ? What can a mixed decimal be changed to? Give an example. But it is convenient in explaining several processes in fractions, to have a method for expressing improper deci. mal fractions, without writing their denominator. The fol. lowing method therefore will be used. Let the inverted separatriz be used to express an im. proper decimal fraction. Thus let the mixed decimal 2,4 which is read two and four tenths, be changed to an im. proper decimal thus, 204 which may be read twenty-four tenths. The denominator af an improper decimal, (like that of other decimals) is always 1 and as many ciphers as there are figures at the right of the separatrix. It is known to be an improper decimal, simply by having its separatrix inverted. Thus 24,69 is read, two thousand four hundred and six. ty-nine hundredths. 239-6 is read, two thousand three hundred and ninety-six tenths, &c. EXAMPLES. Change the following mixed decimals to improper deci. mals, and read them. 246,3 24,96 32,1 326,842 3,6496 49,2643 8,4692 368,491 26,3496 RULE FOR WRITING AN IMPROPER DECIMAL. Write as if the numerator were whole numbers, and place an inverted separatrix, so that there will be as many figures at the right, as there are ciphers in the denominator. Write the following improper decimals. Two hundred and forty-six thousand, four hundred and six tenths. Three millions, five hundred and forty-nine tenths of thousandths. What is the denominator of an improper decimal ? How is it known to be an improper decimal ? What is the rule for writing improper decimals ? Two hundred and sixty-four thousand, five hundred and six thousandths. Five hundred and ninety-six tenths. DECIMAL DIVISION WHEN THE DIVISOR IS A WHOLE NUMBER. The rules for Decimal Division are constructed upon this principle, that any quotient figure must always be put in the same order as the lowest order of that part of the dividend taken. Thus if we divide ,25 (or two tenths, five hundredths,) by 5, the quotient figure must be put in the hundredth or. der, thus, (,05) because the lowest order of the dividend is hundredths. Again, if ,250 is divided by 50, the quotient figure must be 5 thousandths, (,005) for the same reason. Let us then divide ,256 by 2. We proceed exactly as in the Short Division of whole numbers, except in the use of a separatrix. Let the pupil proceed thus : 2),256 ,128 2 tenths divided by 2, gives 1 as quotient, which is 1 tenth, and is set under that order with a separatrix before it. 5 hundredths divided by 2, gives 2 as quotient, which is 2 hundredths, and is set under that order. 1 hundredth remains, which is changed to thousandths, and added to the 6, making 16 thousandths. T'his, divided by 2, gives 8 thousandths as quotient, which is placed in that order. 'If the divisor is a whole number, and has several or. ders in it, we proceed as in Long Division, except we use a separatrix, to keep the figures in their proper order. Thus if we divide 15,12 by 36, we proceed thus : On what principle are the rules for decimal division constructed ? Explain the example given. If the divisor is a whole number, and has se. veral orders, how do we proceed ? 36)15,120,42 14,4 ,72 ,72 ,00 We first take the 15,1, and divide it, remembering that the quotient figure is to be of the same order as the lowest order in the part of the dividend taken, of course the quo. tient 4 is 4 tenths (,4) and must be written thus in the quotient. We now subtract 36 times ,4 which is 14,4, (see rule for Decimal Multiplication, page 108) from the part of the dividend taken and 7 tenths (7) remain. To this bring down the 2 hundredths. Divide, and the quotient figure is 2 hundredths, which must be set in that order in the quotient. Subtract 36 times ,02 (or ,72) from the dividend and nothing remains. Let the following sams be performed and explained as above. Sometimes ciphers must be prefixed to the first quotient figure, to make it stand in its proper order. For example, let ,1512 be divided by 36, and we pro. ceed thus, 36),1512(,0042 ,144 ,0072 0000 Explain the example given. |