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Decimal Fraction

fractions, times and denominator

DECIMAL FRACTION, a fraction whose denominator is a decimal or power of 10. Thus 1234 Ś is a decimal fraction. It may be decom 100 posed into the sum 1000 200 30 4 100 100 100 100 3 4 =10 + 2 + Ś 10 100 By an obvious extension of the method of local values, where each digit has 10 times the value of the like digit, which immediately suc ceeds it, the above decimal fraction may clearly be written more concisely in the form 12.34, where the decimal point after the 2 merely serves to indicate which digit represents units.

In this abbreviated form a decimal fraction is termed a decimal. The operations of addi tion, subtraction, multiplication and division may be applied to decimals in exactly the same man ner as to integers; hence their great utility. They present, nevertheless, this disadvantage, that comparatively few fractional quantities or remainders can be exactly expressed by them; in other words, the greater number of common fractions cannot be reduced, as it is called, to decimal fractions, without leaving a remainder.

Common fractions, such as I, I, 1, and for instance, can be reduced to decimal fractions only by multiplying the numerator and denominator of each by such a number as will convert the denominator into 10, or 100, 1,000, etc. (The common process is merely an abridg

ment to this). But that is possible only when the denominator divides 10 or 100 without re mainder. Thus, of the above denominators, 2 is contained in 10, 5 times; 4 in 100, 25 times; and 25 in 100, 4 times; therefore.

1 1 X 5 5 1 1 X 25 Ś 2 2 X 5 10 4 4 X 25 25 9 9 X 4 36 100 25 25 X 4 100 But neither 3 nor 7 will divide 10 or any power of 10; and therefore these numbers cannot pro duce powers of 10 by multiplication. In such cases we can only approximate the value of the fraction, though this approximation may be made as close as we wish by taking a sufficient number of places after the 0 point. Thus A=3.3333 is less than .0001; 1/4=3.33333 is less than .00001, and so on. Analogues of decimal fractions can be made by taking some other number than 10 as the base of our system of notation: if 6 is our base, for instance, we may write 20 A as 32.143, meaning (3 X 6) -F 2 -F (1 X 1/4) + (4 X 1/4') + (3 X 1/4').