Fractions

FRACTIONS The subject of fractions in arithmetic involves a consideration of the primitive notion of fraction, the growth of the concept, the types of fractions, the symbolism, the names of the parts and the operations. These are discussed in the article on FRACTIONS. So far as elementary arithmetic is concerned, there are three types to be considered : Vulgar fractions (Great Britain) or common fractions (United States), the name referring to such fractions as (a proper fraction), - and -a (both being improper fractions); (2) Sexagesimal fractions, being fractions with unwritten powers of 6o as denominators, as in the case of 27° 1 o' 32", meaning ° and (3) Decimal fractions, generally spoken of as decimals. These types will also be considered in the article on fractions.

Operations with Fractions.

Before decimal fractions began to be used, in the i6th century, and indeed before this use became at all general some two centuries later, the operations with frac tions having large numerators and denominators were very diffi cult. It became desirable to reduce such fractions to lowest terms, a process requiring the finding of the greatest common divisor by the so-called Euclidean method of continued division. Even when this was done, addition and subtraction required the finding of the least common denominator, while multiplication and di vision required still more work in the reduction to lowest terms. With the growing use of decimal fractions the necessity for teach ing the operations with any but the simplest numerical fractions disappeared. Operations with decimals involve no difficulties if they are limited to cases which the pupil will sometime need in daily life.

In Great Britain and to a large extent in the entire British Commonwealth of nations and on the Continent of Europe, the notion of per cent. is used in a somewhat different way from that in the United States. The European usage is the historical one, permitting of such an expression as £6 per cent., meaning a6 out of (or in) £roo. In the United States such an expression is practically never used. The rate of income, expenditure, increase in value or population, and so on is stated abstractly as 6%, Io%, 21%, and the like. Instead of the symbol % being considered to mean "in" or "out of" a hundred, as in L6 in a hundred (pounds), it is looked upon as synonymous with "hundredths" ("hundredth," "of a hundredth") . This usage has led to looking upon the work with per cents. as part of decimals. Since 6% is considered as o.o6, there is no object in treating of the two separately. In any case, however, if the treatment of per cents. is confined to those problems which someone is likely to meet sometime, and is not given largely to impractical puzzles and inverse cases, the sub ject offers no difficulty where measures are decimalized, and but little where they are not.