DECIMAL FRACTIONS 



1731 



DECIMAL FRACTIONS 



.01 



distinguishing, or separating, the fraction from 



the whole, number. Some sign of separation 



was needed. 



These more or 



less clumsy ways 



of designating 



that separation 



have given way 



to the comma 



and decimal 



point. Indeed, 



the decimal point 



was used by one 



writer in 1612, 



but was 

 done d . 



aban- 

 The 

 used 



T 



.08+7 =76 



FIG. 1 



The shaded portion repre- 



comma is used sents 78 units, each of which 

 has the value .01 ; or, 7 units 

 to-day in. many having the value .1, plus 8 



countries of havins the value - 01 - 

 Europe. The United States uses the point. 

 (Mathematicians are interested at present in 

 having the countries agree on one separatrix, 

 America, of course, hoping that the point will 

 be adopted.) With the simple separatrix, the 

 writing of decimal fractions becomes an easy 

 matter. The beginner in decimal fractions 

 writes 1.1, one and 1-tenth; 1.11, one, 1-tenth 

 and 1-hundredth; 16.123, 16 and 1-tenth, 2- 

 hundredths and 3-thousandths. The decimal 

 fractions should be read to him and by him 

 for some time in this way. He is asked to 

 write 4 and 7-tenths and 8-hundredths, and he 

 writes 4.78 and reads it as above; then he is 

 asked to write 4 and 8-hundredths, and, finding 

 he must fill the tenth's place with zero, he 

 writes 4.08, and reads 4 and 8-hundredths. 

 Gradually he reduces the fraction and reads 

 4 and 78-hundredths (see Fig. 1). 



The necessity of this reduction is evident; 

 .78 is 7-tenths and 8-hundredths, which re- 

 duced to hundredths gives 78-hundredths. Ap- 

 proaching it from the other side, 78-hundredths, 

 reduced, gives 7-tenths and 8-hundredths. 



Which is hundredth's place? 

 The second place. 

 Which is thousandth's place? 

 The third place. 



These are the kinds of questions that should 

 come to the beginner in decimal fractions; not 

 "How many places for hundredths? for thou- 

 sandths?" and so on. 



After the reduction, the decimal fraction is 

 read just as a common fraction; the number 

 to the right of the decimal point is the numer- 

 ator, and is read as a whole number, and the 

 denominator is recognized from the last place 



occupied by the numerator; for example, .75 

 is read 75-hundredths, .094 is read 94-thou- 

 sandths, 12.1854 is read 12 and 1854-ten-thou- 

 sandths. These written as common fractions 

 would appear: 



TO~0~> T 0' l^YoTnro" 



Reduction of Decimal Fractions. If .6 is to 

 be considered in hundredths, the hundredth's 

 place is filled with a zero and the fraction 

 appears .60. If it is to be considered as 

 thousandths, both hundredth's and thousandth's 

 places are filled with zeros, and the fraction 

 .700=70=7. 



.600 



.6 = .60 = .600 



Six - tenths = sixty - hundredths = 600 - thou- 

 sandths. 



Other illustrations: 



.75 = .750 .500 - .50 = .5 



75 _ 750 500 50 &_ 



100 1000 1000 = 100 != 10 



From this we see that (1) zeros annexed to 

 a decimal fraction do not change its value, 

 as 7=70=700, and (2) zeros may be dropped 

 from the end of a decimal fraction and the 

 value of the fraction remain the same, as, 

 700=70=7. 



Addition and subtraction of decimal frac- 

 tions hold nothing new for the child if he 

 has appreciated the place value idea set forth 

 above. He adds or subtracts just as in whole 

 numbers; no time need be given to these. He 

 sees that he needs the decimal point in the 

 answer between units and tenths just as in the 

 addends or above the line. He knows that 

 "the points must come under each other" so 

 that tenths are all in the same "place," or 

 column, hundredths all in the same place, etc. 

 So, without preliminaries he adds or subtracts 

 as follows, just as in whole numbers: 



7.8 



16.48 14.693 



12.096 - 8.452 ' 



36.376 6.241 



124-16.943 = n 

 124.000 

 - 16.943 

 107.057 



7.8 7.800 



4.586= 4.586 



19.81 19.810 



32.196 



The numbers to be added or subtracted may 



be reduced to the same denomination as in 



the last two problems above, by filling the 



places with zeros. 



After the child has worked a day or two 



