(43.) A decimal fraction is one which has for its denominator an unit, with as many cyphers annexed to it as the numerator has places of figures. It is customary to write the numerator only, since by the number of places it contains, the value of the denominator is known. Thus, is written,4,1,24 and,361. When the numerator does not contain as many figures as the denominator has cyphers, the deficiency is supplied by writing ciphers at the left hand; thus, T음이 is written,05 음이,032, and ᅮ이이이의,0011.

A mixed number, made up of a whole number and a decimal fraction, is written with a point called the decimal point, between the whole number and the fractional part; thus, 4,67 is the same as 4, or

10

6 100

Ciphers on the right hand of a decimal do not alter its value, for, and, are the same, since they can each be reduced to. But ciphers on the left hand do alter the value of the decimal; while, 3 represents 주이,, 03 is but 름이, and ,003 is but 이이이

From the method of notation adopted in decimal fractions, it is evident that decimals, like whole numbers, decrease from left to right in a tenfold proportion, each place, as we recede from the left, having but one tenth the value of that which precedes it. Thus, if we commence at the left in whole numbers, we find each place representing a unit of a lower order, till we arrive at the lowest, namely, units. Continuing the same law of decrease, the first figure to the right of units becomes 10th parts, the second, 100th parts, the third, 1000th parts, and so on.

We have then only to apply the system of numeration for whole numbers, counting from the decimal point both ways;-towards the left, units, tens, hundreds, &c.; and towards the right, tenth parts, hundredth parts, thousandth parts, &c., as is represented in the following table :

The expression in the table is read, 2 millions, 367 thou

sands, 431, and 347 thousands, 632 millionths.

(44.) Decimal fractions result from the division of one number by another in the following manner :

Let it be required to divide 63 by 8.

8)63(7,875.

56

70

64

60

56

40

40

I find 8 contained 7 times in 63, with a remainder of 7; I now conceive that each of the units in 7 is divided into 10 parts, which gives me 70 tenth parts of a unit for a new dividend; this is divisible by 8; and since this second dividend is tenth parts, the quotient will be tenth parts; and hence the 8 in the quotient is written with the decimal point before it. I now find a remainder of 6 tenth parts of a unit; each of these parts are also subdivided into ten parts, or, which is the same thing, this remainder is multiplied by 10, which gives 60 hundredth parts; the quotient of which, by the divisor, is 7 hundredths, which is put in the place of hundredths in the quotient. The next remainder is likewise multiplied by 10, which gives thousandth parts for a new dividend, and a figure in the place of thousandths, in the quotient.

Hence it appears, that a division may be executed in decimals, when it cannot be in whole numbers. In the preceding example, the quotient of 63 by 8 is 7,875. Expressed by a whole number and vulgar fraction, it would have been 77. By inspecting the work, it will be seen, that the vulgar fraction, is reduced to a decimal by adding cyphers to the numerator (for this is multiplying by 10), and dividing by the denominator. The following rule will now require no explanation.

To reduce a vulgar fraction to a decimal.
RULE.

(45.) Annex ciphers to the numerator, and divide by the denominator, and point off as many places for decimals in the quotient, as there are ciphers annexed to the

numerator.

(46.) Write down the decimal with its denominator, and then reduce it to its lowest terms.

(47.) Circulating decimals are those, in which the same figures are repeated at regular intervals in the same order; ,324324, &c., and,5454, &c. are circulating decimals. When a single figure is repeated as,,3333, the decimal is sometimes called a repeating decimal. Circulating decimals result from divisions in which the divisor and dividend are prime to each other, and the divisor is such a number, that it is not contained without remainder, in any number expressed by 1, and any number of ciphers, as 10, 100, 1000, &c. Take, for example, the fraction. If we attempt to reduce this fraction to a decimal by the preceding rules, we shall find the same numbers recurring in the remainders; and hence the same numbers will be repeated in the quotient.

EXAMPLE.

11)60(0,5454
55

50

44

60

55

50

44

60

The numbers repeated are called periods. In this example two figures form the period.

It is evident, that the true value of this fraction cannot be expressed in a decimal; and it is sometimes convenient, when such a decimal fraction occurs, to change it into its equivalent vulgar fraction. This may be done by the following

RULE.

Write under the figures which constitute one period, as many 9s as there are places of figures in the period. This will form the fraction from which the circulating decimal was derived.

For another example, reduce 324324324 to a vulgar

fraction.

32

Ans. = 36