2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 Thus, we perceive that 3 in the units' place has a value of 3 units; moved one place to the left, its value is 10×3 or 30; and moved one place to the right its value is one-tenth of 3, or 3 tenths, and so on. 169. If it were required to write one-tenth, without integer, the expression would be .1; one hundredth, .01; one thousandth, .001, &c. Six-tenths would be .6; five hundredths, .05; eight thousandths, .008, &c. 170. Read and write down the following quantities: 3.6, 50.24, 36.07, 19.004, 31.042, 124.103, 95.0003, .6007, 1.011, 7.00042, 17.200042, 5.00024, 95.0000241, 7.07081. 40 171. Now, let us notice what alteration takes place in the value of a decimal, when one or more ciphers are added to it, thus: .4, .40, .400, &c. If these fractions be expressed as common fractions, we have; b, 1%, 100%, which are evidently of the same value, Therefore, ciphers annexed to the right of a decimal have no effect upon its value. Then, .5 .50 .500, &c.; .26 .260; .005.0050=.00500, &c. = 172. These properties of decimals show us at once their advantages over vulgar fractions, for the tedious process of reduction to the same denominator is here done away with, since decimals which have the same number of figures, have the same denominator. The following examples will illustrate this:5.9, 4.24, 7.0072, are the same as 5.9000, 4.2400, 7.0072, 173. If ciphers be prefixed to the left of a decimal, after the point, what effect has it on its value? Let .7, .07, .007, &c., the fractions. be We know that .7%; .07=130, .007=Tooo, &c.; each fraction being one-tenth part of that which immediately precedes it. Therefore, every cipher affixed to the left hand of a decimal, after the point, decreases its value ten-fold. 174. This property leads us to a brief method of multiplying and dividing decimals by 10, 100, 1000, &c. Take, for instance, 24.345, and let the point be moved two places to the right, we have 2434.5. Now, 24.345 = 24345, and 2434.5=24345 ; hence, we see that the second fraction has 100 times the value of the first, because its denominator is 100 times smaller. 10 175. On the other hand, let the point be moved one place to the left, then 24.345 becomes 2.4345; and, since 24.345= 2345, and 2.43451338, we perceive that the second fraction has a denominator 10 times larger than the first; therefore, the last fraction is 10 times smaller. Hence, a decimal is multiplied by 10, 100, 1000, &c., if the point be shifted one, two, three, &c., places towards the right; and, conversely, a decimal is divided by 10, 100, 1000, &c., if the point be shifted one, two, three, &c., places towards the left. 34.63 x 10-346.3; 1.14 × 100=114; 63.2304 x 1000=63230.4. 241.63÷10=24.163; 354.2÷100 =3.542; 763.9÷1000.7639. 176. To ascertain the greater of two decimal fractions, it is not the number of figures which must be considered, but their magnitude and position. Thus, .4<.51; 7>.5432; .005> .00087; .09<.1; .548>.5437. 177. Suppose it were necessary to express a vulgar fraction by a decimal. In every fraction the numerator may be considered as the dividend, and the denominator as the divisor; for instance, signifies 3 divided by 5. If we affix decimal ciphers to the numerator, we have: 30390.6; for 5 not being contained in 3 units, is contained in 3.0 or 30-tenths, 6-tenths or .6. To reduce to a decimal fraction : 8)50(.625 20 16 40 In this example, 8 is not contained in 5, but it is contained in 50 tenths, 6 tenths and 2 over, which is altered into 20 hundredths by the addition of a cipher; 8 is contained in 20 hundredths, 2 hundredths, with the remainder 4, which becomes 40 thousandths by adding one cipher, and 8 in 40 thousandths gives 5 thousandths times exactly; therefore, 5·000=.625. Hence, to reduce a vulgar fraction to a decimal, divide the numerator with as many ciphers added to the right of it, as are necessary, by the denominator, and the quotient will consist of as many decimal places as there are ciphers added. 178. Transform the vulgar fractions ; ; 13; 128; 2 to decimals. 179. It sometimes happens that the division will not terminate, but the same figures will recur over again perpetually. &c. Ex. 35=75÷98.333, &c.; = .4285714, &c.; s=.2777, Decimals of this kind, in which the figures are continually repeated in some certain order, are called recurring, repeating, or circulating decimals; the part repeated is termed the period or repetend. The circulating decimal is pure when the period begins immediately after the point, and mixed if it consists of a non-recurring and a recurring part. The period is distinguished by points placed over the first and last figures of the repetend, and the above results become: 83; = .428571; †=.27. 180. The operation of determining a long period of a decimal whose denominator is large, is shortened by the following process:: Let be the fraction to be converted into a decimal. = By division we find that .05263+; again,9x.0526315789.4736842105%; and, therefore, .05263157894736842105%; = hence,.42105263157894736842. Then the period comprises 18 figures. It will have been observed, that throughout the process, the period consists of the same figures, whatever the numerator of the fraction is, though they do not begin with the same figure. Express as decimals: of; of; of 1 ; % 11⁄2 11; 181. EXERCISES. ; 16 ;; √6; 37; 254; 14; 21; 77%; } ; ; ; 18; 37; } }; }; 3; 4; 1735. ៖ 182. Let it be required to reduce a terminating decimal to a vulgar fraction. Since the decimal is the numerator of a common fraction, whose denominator is 1, to which as many ciphers are added as there are digits after the point. Therefore: Τ .5==; .75=75%=1; .048=1880-195; 54.6=54.6 = 273. 183. EXERCISES. Convert the following decimals into vulgar fractions: .125; .30; .65; .72; .945; .97; .705; .075; 7.245; 9.027; 16.00075; 18.1642. 184. It has been explained how to reduce any vulgar fraction to a decimal; the converse is, therefore, true. Any decimals, whether pure or mixed recurring, can be represented by a vulgar fraction; we shall now determine a method to find the exact value of a circulating decimal. First, of a pure recurring decimal: By dividing, we find that =.1111, &c.=.i; hence, .i is the decimal equivalent to Also, #=.2222 &c.=.2; hence, .2 is the decimal equivalent to 3. =.3333 &c.=.3 =&c. .3 Hence, every pure recurring decimal, having one figure in the period, is equivalent to a common fraction, whose numerator is that figure and the denominator 9. Again, .010101 &c.=.01; consequently,=.0202 &c. =02; 88=56; 73=.74. Hence every pure recurring decimal, having two figures in the period, is equivalent to a common fraction, whose numerator is composed of those figures, and the denominator 99. = Likewise, =.001001 &c.=.001, and 351.754 &c. Similarly,1.00010001 &c. = .0001, and 33§§=‚2356 &c. Hence it follows that any pure circulating decimal is equivalent to a vulgar fraction whose numerator is the period, and denominator as many 9's as there are figures in the period. Thus: 963-888=497; 1569=1888-23: 000729= vb°L?%%=3,°37 729 185. Secondly, when the recurring decimal is mixed. Such a decimal may always be decomposed into the sum of two vulgar fractions. 8x9+5 8(10-1)+5_80+5-8 90 Therefore, .85 = 33. 90 In the same manner, .8367=3+, § z o 83(100-1)+67 8300+67-83 9 = 90 83 × 99+67 9900 From the consideration of the preceding examples, we infer the following law : To find the value of a mixed recurring decimal, take for the numerator the difference between the number expressed by the mixed decimal, and that expressed by the non-recurring part, and for the denominator as many nines as there are recurring figures followed by as many ciphers as there are non-recurring figures. 186. Another method, for the reduction of a recurring decimal into an equivalent vulgar fraction, which is well adapted to practice, is the following: Let be the decimal; any symbol x may be used to represent the value required. And subtracting the former from the latter, we have :—— 10x-x=7.7777 &c. - .7777 &c. ; or, 9x=7; therefore, x=3}. |