« ΠροηγούμενηΣυνέχεια »
indicated by the horizontal line, while a decimal fraction may be changed into a common one by writing under it its secondary name in figures, instead of denoting it by a separatrix. Thus, ='6; and 6=%= This definition gives us the eleventh principle of arithmetic, as follows:
XI. An integer or a decimal fraction is changed into a com
mon fraction by expressing its secondary name under it in figures ; a common fraction is changed to a decimal one, or to an integer, by performing the division indicated.
3. Denominate fractions are subdivisions of measures of any kind, whether of length, surface, or solidity; of weights, money, time, &c. In calculations, their secondary name is written over or beside them, in words, contracted or in full, as gr., or grains, p., or pecks, or in conventional characters, as 3, 3, for drams and ounces. The unit in this kind of fractions is a certain conventional measure, such as pound, bushel, yard, dollar, year, &c., of which the fractions are subdivisions.
Questions by the Teacher. -How many kinds of fractions are there ? Name them. What is the primary name of all numbers? The same as that of Whence is the secondary name of integers and decimal fractions derived ? Why is this sometimes called their local name? How is the secondary name of common fractions ascertained? What is the meaning of the word vulgar, frequently used in place of common fractions ? Why is the word common preferable? Ans. Because the word vulgar is now chiefly used in the sense of mean, rude, low. How is the secondary name of denominate fractions ascertained? What is the unit in denominate fractions ?
SECTION I.-Common Fractions.
[For a full development of the first principles of Common Fractions, see Oral Arithmetic, Chap. II. and III. throughout.]
I. Change of Form. Remarks and Definitions.- Common fractions are capable of assuming, as already noticed, an infinite variety of forms. For, as no change of value takes place when both terms are multiplied by the same number, it is evident that by multiplication alone they may be infinitely varied. Thus, i assumes the
forms of \, , 186, and so on without end, by continual multiplication of both terms by 2; and the same fraction, ), or any other, will assume other endless series of forms by multiplying both terms by 3, 4, 5, 6, or any other number, and all this without the slightest change of value. The same remark applies also to integers and to decimal and denominate fractions, since each of these may assume the form of a common fraction, simply by writing their secondary value under them in figures. Thus 6, 18, 625, 4s. (shillings), may take the fractional forms of f, x, y, 4s., a shape in which they are susceptible of all the variety of form shown above. This capacity of change of form is exceedingly useful in simplifying and abridging the operations of arithmetic. There are six kinds of common fractions, which may
be arranged under two heads : 1, proper, improper, and mixed; 2, simple, compound, and complex.
1. A proper fraction is less than unity, and consequently its numerator is less than its denominator, as . An improper fraction is greater than unity, consequently its numerator is greater than its denominator, as or 1. When an integer is separated from the fractional part by division, the improper fraction becomes a mixed number. Thus, the improper fractions 5, 1, are the same as the mixed numbers 14, 37, the integers being separated in the latter by the partial performance of the division indicated. Integers in a fractional form, without accompanying fractions, such as f, or 12, are also considered improper fractions.
simple fraction is a fraction in a single expression. It may either be proper, as ļ, or improper, as . A compound fraction is composed of two or more expressions. It is a fraction of a fraction, or a fraction of an integer, and is known by the word of between the expressions, a word which was found in Oral Arithmetic, p. 83, 13, always to indicate multiplication in this connection. Thus, of Į, or of of or 4 of 3, are compound fractions. By performing the multiplication indicated, they become simple; the first being tá, the second , and the third q. A complex fraction is a fraction in which one or both terms are themselves fractional, as
44 31 6
12' 57' 153' 을
These, also, become simple by performing the operations } indicated. Any pupil that is familiar with Chap. III., Oral
Arithmetic, can change their forms to those of simple fractions by the aid of the following questions : First, 41. How many fourths in 1? How many in 4 ?
12 In 41 ? How much is divided by 12 ? Second,
How many fifths in 37? How many fourths
51 in 51? Divide Le by 4. Third,
Put 6 in a fractional form. How many
155. eighths in 159 ? Divide by 192. Fourth,
Divide by }, as indicated by the horizontal line.
a. To change a fraction to its equivalent lowest expression.
1. Change is to its lowest expression, by striking out the prime factors common to both terms.
28 2 • 2.7
35 5.7 Suggestive Questions.—What factors are the same in both terms ? What will the fraction be if the factor 7 be omitted in both terms? Is the lowest term of the fraction ?
2. Change 475 to its lowest expression.
Suggestwe Question. When the factors common to both terms are stricken out, what will be the lowest expression of the fraction ?
3. Change 1996 to its lowest expression by inspection merely.
4. Change 25to its lowest expression by inspection.
b. To change an integer or a decimal fraction to the form of
an equivalent common fraction. 1. Change 42 to the form of an equivalent common fraction. See p. 193, XI.
2. Change 3-75 to an equivalent common fraction.
3. Change 7.07, 245, 003, 4.25, 265, severally to the form of equivalent common fractions.
C. To change a common fraction, whether proper or improper,
to a whole or mixed number, or to a decimal fraction. 1. Change ito, v 18, 125, severally, to decimal fractions, or mixed numbers; that is, perform the division indicated, and prove by rechange to common fractions. See 11th principle,
2. Change 200, 450, and , severally, to whole numbers, and prove by rechange to common fractions.
3. Change 1, 5, 1, 15, and 4, severally, to decimal fractions, and prove by rechange to their original form.
4. Change 1, 2, 3, and 45, severally, to mixed numbers, and prove by rechange to their original form.
I. Every decimal fraction consists of a specified number of tenths, or of some power of tenths, such as hundredths, thousandths, &c.; consequently every fraction that cannot be expressed in tenths, or one of the powers of tenths, cannot be accurately expressed in a decimal form. It follows, then, that in changing common fractions to a decimal form, the division will never terminate, but go on to infinity in every case where any prime factor other than 2 or 5 (prime factors of 10) remains in the denominator after the fraction has been brought to its lowest terms. For instance, if 3 or 7 (or any other prime factor but 2 and 5) should be a factor in the denomi. nator and not in the numerator, an undivided remainder would always occur in the division, and, by adding a cipher, the quotient would form an endless series of figures. Thus, if we perform the division indicated in }, it gives -333, &c., without end; and in it gives •324324324, &c., a period of three figures endlessly recurring. Such decimals as these, which continually recur in periods of one or more figures in the same order, are called circulating decimals. They are distinguished
from ordinary decimals by a dot placed over the first and last figure of the circulating period. Thus, į is expressed by 3, and 14 by 324. The set of figures which repeats is called a repetend.
II. When a period begins with the first decimal figure, it is called a simple repetend. But when other decimal figures occur before the period commences, it is called a compound repetend. Thus, į='333, &c., forms a simple, and £=*1666, &c., forms a compound repetend.
1. Decimal Fractions with Simple Repetends. 1. Change the following common fractions to equivalent decimal fractions carried to twelve or fifteen places, allowing the operation to remain on the slate till examined by the following questions :
&c. 2. g=
&c. 4. &c.
&c. Suggestive Questions.—What decimal fraction is equivalent to j? Then what decimal fraction is equivalent to f, or 2 times f? What to f? To $? &c., up to f? If, then, it were required to change a decimal fraction of one figure continually repeated, that is, a circulating decimal with one figure for repetend, into a common fraction, what would be the numerator ? What the denominator ?
2. What decimal fraction is equivalent to o'y? What to og, or 2 times g'? To ong? To ? To go? To jo? To
? To ? &c. If, then, it be required to change a circulating decimal with two figures for a repetend into a common fraction, what will be the numerator ? What the denominator ?
3. What decimal fraction is equivalent to ggg? What to ŠT? To ogg? Togg? &c. To To je? To 693? To $77? &c. ? If, then, it be required to change a circulating decimal with three figures for a repetend into a common fraction, what would be the numerator ? What the denominator ?
4. In general, then, if it be required to change a circulating decimal with any number of figures for a repetend, what would be the numerator? What the denominator ? May it not, then, be considered as the 12th principle of arithmetic, that,